Aljabar Linear Contoh

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}]$

Langkah 1

Temukan nilai eigennya.

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Langkah 1.1

Gunakan rumus untuk menentukan persamaan karakteristik $p (λ)$ .

$p (λ) = determinan (A - λ I_{4})$

Langkah 1.2

Matriks satuan atau matriks satuan dengan ordo $4$ adalah matriks persegi $4 \times 4$ dengan bilangan satu di diagonal utama dan nol di elemen lainnya.

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Langkah 1.3

Substitusikan nilai-nilai yang diketahui ke dalam $p (λ) = determinan (A - λ I_{4})$ .

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Langkah 1.3.1

Substitusikan $[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}]$ untuk $A$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] - λ I_{4})$

Langkah 1.3.2

Substitusikan $[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$ untuk $I_{4}$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Langkah 1.4

Sederhanakan.

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Langkah 1.4.1

Sederhanakan setiap suku.

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Langkah 1.4.1.1

Kalikan $- λ$ dengan setiap elemen di dalam matriks tersebut.

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2

Sederhanakan setiap elemen dalam matriks.

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Langkah 1.4.1.2.1

Kalikan $- 1$ dengan $1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.2

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.2.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.2.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.3

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.3.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.3.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.4

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.4.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.4.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.5

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.5.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.5.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.6

Kalikan $- 1$ dengan $1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.7

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.7.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.7.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.8

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.8.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.8.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.9

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.9.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.9.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.10

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.10.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.10.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.11

Kalikan $- 1$ dengan $1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.12

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.12.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.12.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.13

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.13.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.13.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.14

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.14.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.14.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.15

Kalikan $- λ \cdot 0$ .

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Langkah 1.4.1.2.15.1

Kalikan $0$ dengan $- 1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 λ & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.15.2

Kalikan $0$ dengan $λ$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \cdot 1 \end{array}])$

Langkah 1.4.1.2.16

Kalikan $- 1$ dengan $1$ .

$p (λ) = determinan ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}])$

Langkah 1.4.2

Tambahkan elemen yang seletak.

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - λ & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3

Simplify each element.

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Langkah 1.4.3.1

Tambahkan $2$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - λ & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.2

Tambahkan $6$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 + 0 \\ 1 + 0 & 56 - λ & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.3

Tambahkan $8$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 + 0 & 56 - λ & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.4

Tambahkan $1$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.5

Tambahkan $2$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.6

Tambahkan $85$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 + 0 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.7

Tambahkan $16$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 & 7 + 0 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.8

Tambahkan $7$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 & 7 & 9 - λ & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.9

Tambahkan $3$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 & 7 & 9 - λ & 3 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.10

Tambahkan $4$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 & 7 & 9 - λ & 3 \\ 4 & 5 + 0 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.11

Tambahkan $5$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 & 7 & 9 - λ & 3 \\ 4 & 5 & 7 + 0 & 99 - λ \end{array}]$

Langkah 1.4.3.12

Tambahkan $7$ dan $0$ .

$p (λ) = determinan [\begin{array}{c} 1 - λ & 2 & 6 & 8 \\ 1 & 56 - λ & 2 & 85 \\ 16 & 7 & 9 - λ & 3 \\ 4 & 5 & 7 & 99 - λ \end{array}]$

Langkah 1.5

Find the determinant.

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Langkah 1.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Langkah 1.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Langkah 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Langkah 1.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 56 - λ & 2 & 85 \\ 7 & 9 - λ & 3 \\ 5 & 7 & 99 - λ \end{array} |$

Langkah 1.5.1.4

Multiply element $a_{11}$ by its cofactor.

$(1 - λ) | \begin{array}{c} 56 - λ & 2 & 85 \\ 7 & 9 - λ & 3 \\ 5 & 7 & 99 - λ \end{array} |$

Langkah 1.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} |$

Langkah 1.5.1.6

Multiply element $a_{12}$ by its cofactor.

$- 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} |$

Langkah 1.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} |$

Langkah 1.5.1.8

Multiply element $a_{13}$ by its cofactor.

$6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} |$

Langkah 1.5.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.1.10

Multiply element $a_{14}$ by its cofactor.

$- 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.1.11

Add the terms together.

$p (λ) = (1 - λ) | \begin{array}{c} 56 - λ & 2 & 85 \\ 7 & 9 - λ & 3 \\ 5 & 7 & 99 - λ \end{array} | - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2

Evaluasi $| \begin{array}{c} 56 - λ & 2 & 85 \\ 7 & 9 - λ & 3 \\ 5 & 7 & 99 - λ \end{array} |$ .

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Langkah 1.5.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Langkah 1.5.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Langkah 1.5.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Langkah 1.5.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 9 - λ & 3 \\ 7 & 99 - λ \end{array} |$

Langkah 1.5.2.1.4

Multiply element $a_{11}$ by its cofactor.

$(56 - λ) | \begin{array}{c} 9 - λ & 3 \\ 7 & 99 - λ \end{array} |$

Langkah 1.5.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} |$

Langkah 1.5.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} |$

Langkah 1.5.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |$

Langkah 1.5.2.1.8

Multiply element $a_{13}$ by its cofactor.

$85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |$

Langkah 1.5.2.1.9

Add the terms together.

$p (λ) = (1 - λ) ((56 - λ) | \begin{array}{c} 9 - λ & 3 \\ 7 & 99 - λ \end{array} | - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2

Evaluasi $| \begin{array}{c} 9 - λ & 3 \\ 7 & 99 - λ \end{array} |$ .

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Langkah 1.5.2.2.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) ((56 - λ) ((9 - λ) (99 - λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2

Sederhanakan determinannya.

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Langkah 1.5.2.2.2.1

Sederhanakan setiap suku.

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Langkah 1.5.2.2.2.1.1

Perluas $(9 - λ) (99 - λ)$ menggunakan Metode FOIL.

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Langkah 1.5.2.2.2.1.1.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) ((56 - λ) (9 (99 - λ) - λ (99 - λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.1.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) ((56 - λ) (9 \cdot 99 + 9 (- λ) - λ (99 - λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.1.3

Terapkan sifat distributif.

$p (λ) = (1 - λ) ((56 - λ) (9 \cdot 99 + 9 (- λ) - λ \cdot 99 - λ (- λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2

Sederhanakan dan gabungkan suku-suku sejenis.

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Langkah 1.5.2.2.2.1.2.1

Sederhanakan setiap suku.

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Langkah 1.5.2.2.2.1.2.1.1

Kalikan $9$ dengan $99$ .

$p (λ) = (1 - λ) ((56 - λ) (891 + 9 (- λ) - λ \cdot 99 - λ (- λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.2

Kalikan $- 1$ dengan $9$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - λ \cdot 99 - λ (- λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.3

Kalikan $99$ dengan $- 1$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - 99 λ - λ (- λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.4

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - 99 λ - 1 \cdot - 1 λ \cdot λ - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.5

Kalikan $λ$ dengan $λ$ dengan menambahkan eksponennya.

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Langkah 1.5.2.2.2.1.2.1.5.1

Pindahkan $λ$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - 99 λ - 1 \cdot - 1 (λ \cdot λ) - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.5.2

Kalikan $λ$ dengan $λ$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - 99 λ - 1 \cdot - 1 λ^{2} - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.6

Kalikan $- 1$ dengan $- 1$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - 99 λ + 1 λ^{2} - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.1.7

Kalikan $λ^{2}$ dengan $1$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 9 λ - 99 λ + λ^{2} - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.2.2

Kurangi $99 λ$ dengan $- 9 λ$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 108 λ + λ^{2} - 7 \cdot 3) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.1.3

Kalikan $- 7$ dengan $3$ .

$p (λ) = (1 - λ) ((56 - λ) (891 - 108 λ + λ^{2} - 21) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.2

Kurangi $21$ dengan $891$ .

$p (λ) = (1 - λ) ((56 - λ) (- 108 λ + λ^{2} + 870) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.2.2.3

Susun kembali $- 108 λ$ dan $λ^{2}$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.3

Evaluasi $| \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} |$ .

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Langkah 1.5.2.3.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (7 (99 - λ) - 5 \cdot 3) + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.3.2

Sederhanakan determinannya.

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Langkah 1.5.2.3.2.1

Sederhanakan setiap suku.

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Langkah 1.5.2.3.2.1.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (7 \cdot 99 + 7 (- λ) - 5 \cdot 3) + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.3.2.1.2

Kalikan $7$ dengan $99$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (693 + 7 (- λ) - 5 \cdot 3) + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.3.2.1.3

Kalikan $- 1$ dengan $7$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (693 - 7 λ - 5 \cdot 3) + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.3.2.1.4

Kalikan $- 5$ dengan $3$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (693 - 7 λ - 15) + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.3.2.2

Kurangi $15$ dengan $693$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4

Evaluasi $| \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |$ .

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Langkah 1.5.2.4.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (7 \cdot 7 - 5 (9 - λ))) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4.2

Sederhanakan determinannya.

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Langkah 1.5.2.4.2.1

Sederhanakan setiap suku.

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Langkah 1.5.2.4.2.1.1

Kalikan $7$ dengan $7$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (49 - 5 (9 - λ))) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4.2.1.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (49 - 5 \cdot 9 - 5 (- λ))) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4.2.1.3

Kalikan $- 5$ dengan $9$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (49 - 45 - 5 (- λ))) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4.2.1.4

Kalikan $- 1$ dengan $- 5$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (49 - 45 + 5 λ)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4.2.2

Kurangi $45$ dengan $49$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (4 + 5 λ)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.4.2.3

Susun kembali $4$ dan $5 λ$ .

$p (λ) = (1 - λ) ((56 - λ) (λ^{2} - 108 λ + 870) - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5

Sederhanakan determinannya.

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Langkah 1.5.2.5.1

Sederhanakan setiap suku.

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Langkah 1.5.2.5.1.1

Perluas $(56 - λ) (λ^{2} - 108 λ + 870)$ dengan mengalikan setiap suku dalam pernyataan pertama dengan setiap suku dalam pernyataan kedua.

$p (λ) = (1 - λ) (56 λ^{2} + 56 (- 108 λ) + 56 \cdot 870 - λ \cdot λ^{2} - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2

Sederhanakan setiap suku.

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Langkah 1.5.2.5.1.2.1

Kalikan $- 108$ dengan $56$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 56 \cdot 870 - λ \cdot λ^{2} - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.2

Kalikan $56$ dengan $870$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ \cdot λ^{2} - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.3

Kalikan $λ$ dengan $λ^{2}$ dengan menambahkan eksponennya.

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Langkah 1.5.2.5.1.2.3.1

Pindahkan $λ^{2}$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - (λ^{2} λ) - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.3.2

Kalikan $λ^{2}$ dengan $λ$ .

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Langkah 1.5.2.5.1.2.3.2.1

Naikkan $λ$ menjadi pangkat $1$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - (λ^{2} λ^{1}) - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.3.2.2

Gunakan kaidah pangkat $a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{2 + 1} - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.3.3

Tambahkan $2$ dan $1$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{3} - λ (- 108 λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.4

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{3} - 1 \cdot - 108 λ \cdot λ - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.5

Kalikan $λ$ dengan $λ$ dengan menambahkan eksponennya.

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Langkah 1.5.2.5.1.2.5.1

Pindahkan $λ$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{3} - 1 \cdot - 108 (λ \cdot λ) - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.5.2

Kalikan $λ$ dengan $λ$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{3} - 1 \cdot - 108 λ^{2} - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.6

Kalikan $- 1$ dengan $- 108$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{3} + 108 λ^{2} - λ \cdot 870 - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.2.7

Kalikan $870$ dengan $- 1$ .

$p (λ) = (1 - λ) (56 λ^{2} - 6048 λ + 48720 - λ^{3} + 108 λ^{2} - 870 λ - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.3

Tambahkan $56 λ^{2}$ dan $108 λ^{2}$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6048 λ + 48720 - λ^{3} - 870 λ - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.4

Kurangi $870 λ$ dengan $- 6048 λ$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} - 2 (- 7 λ + 678) + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.5

Terapkan sifat distributif.

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} - 2 (- 7 λ) - 2 \cdot 678 + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.6

Kalikan $- 7$ dengan $- 2$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} + 14 λ - 2 \cdot 678 + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.7

Kalikan $- 2$ dengan $678$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} + 14 λ - 1356 + 85 (5 λ + 4)) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.8

Terapkan sifat distributif.

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} + 14 λ - 1356 + 85 (5 λ) + 85 \cdot 4) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.9

Kalikan $5$ dengan $85$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} + 14 λ - 1356 + 425 λ + 85 \cdot 4) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.1.10

Kalikan $85$ dengan $4$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6918 λ + 48720 - λ^{3} + 14 λ - 1356 + 425 λ + 340) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.2

Tambahkan $- 6918 λ$ dan $14 λ$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6904 λ + 48720 - λ^{3} - 1356 + 425 λ + 340) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.3

Tambahkan $- 6904 λ$ dan $425 λ$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6479 λ + 48720 - λ^{3} - 1356 + 340) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.4

Kurangi $1356$ dengan $48720$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6479 λ - λ^{3} + 47364 + 340) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.5

Tambahkan $47364$ dan $340$ .

$p (λ) = (1 - λ) (164 λ^{2} - 6479 λ - λ^{3} + 47704) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.6

Pindahkan $- 6479 λ$ .

$p (λ) = (1 - λ) (164 λ^{2} - λ^{3} - 6479 λ + 47704) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.2.5.7

Susun kembali $164 λ^{2}$ dan $- λ^{3}$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 | \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} | + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3

Evaluasi $| \begin{array}{c} 1 & 2 & 85 \\ 16 & 9 - λ & 3 \\ 4 & 7 & 99 - λ \end{array} |$ .

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Langkah 1.5.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $2$ by its cofactor and add.

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Langkah 1.5.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Langkah 1.5.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Langkah 1.5.3.1.3

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 2 & 85 \\ 7 & 99 - λ \end{array} |$

Langkah 1.5.3.1.4

Multiply element $a_{21}$ by its cofactor.

$- 16 | \begin{array}{c} 2 & 85 \\ 7 & 99 - λ \end{array} |$

Langkah 1.5.3.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} |$

Langkah 1.5.3.1.6

Multiply element $a_{22}$ by its cofactor.

$(9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} |$

Langkah 1.5.3.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |$

Langkah 1.5.3.1.8

Multiply element $a_{23}$ by its cofactor.

$- 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |$

Langkah 1.5.3.1.9

Add the terms together.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 | \begin{array}{c} 2 & 85 \\ 7 & 99 - λ \end{array} | + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.2

Evaluasi $| \begin{array}{c} 2 & 85 \\ 7 & 99 - λ \end{array} |$ .

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Langkah 1.5.3.2.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (2 (99 - λ) - 7 \cdot 85) + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.2.2

Sederhanakan determinannya.

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Langkah 1.5.3.2.2.1

Sederhanakan setiap suku.

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Langkah 1.5.3.2.2.1.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (2 \cdot 99 + 2 (- λ) - 7 \cdot 85) + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.2.2.1.2

Kalikan $2$ dengan $99$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (198 + 2 (- λ) - 7 \cdot 85) + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.2.2.1.3

Kalikan $- 1$ dengan $2$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (198 - 2 λ - 7 \cdot 85) + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.2.2.1.4

Kalikan $- 7$ dengan $85$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (198 - 2 λ - 595) + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.2.2.2

Kurangi $595$ dengan $198$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) | \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.3

Evaluasi $| \begin{array}{c} 1 & 85 \\ 4 & 99 - λ \end{array} |$ .

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Langkah 1.5.3.3.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (1 (99 - λ) - 4 \cdot 85) - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.3.2

Sederhanakan determinannya.

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Langkah 1.5.3.3.2.1

Sederhanakan setiap suku.

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Langkah 1.5.3.3.2.1.1

Kalikan $99 - λ$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (99 - λ - 4 \cdot 85) - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.3.2.1.2

Kalikan $- 4$ dengan $85$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (99 - λ - 340) - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.3.2.2

Kurangi $340$ dengan $99$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (- λ - 241) - 3 | \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.4

Evaluasi $| \begin{array}{c} 1 & 2 \\ 4 & 7 \end{array} |$ .

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Langkah 1.5.3.4.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (- λ - 241) - 3 (1 \cdot 7 - 4 \cdot 2)) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.4.2

Sederhanakan determinannya.

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Langkah 1.5.3.4.2.1

Sederhanakan setiap suku.

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Langkah 1.5.3.4.2.1.1

Kalikan $7$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (- λ - 241) - 3 (7 - 4 \cdot 2)) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.4.2.1.2

Kalikan $- 4$ dengan $2$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (- λ - 241) - 3 (7 - 8)) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.4.2.2

Kurangi $8$ dengan $7$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ - 397) + (9 - λ) (- λ - 241) - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5

Sederhanakan determinannya.

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Langkah 1.5.3.5.1

Sederhanakan setiap suku.

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Langkah 1.5.3.5.1.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (- 16 (- 2 λ) - 16 \cdot - 397 + (9 - λ) (- λ - 241) - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.2

Kalikan $- 2$ dengan $- 16$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ - 16 \cdot - 397 + (9 - λ) (- λ - 241) - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.3

Kalikan $- 16$ dengan $- 397$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 + (9 - λ) (- λ - 241) - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.4

Perluas $(9 - λ) (- λ - 241)$ menggunakan Metode FOIL.

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Langkah 1.5.3.5.1.4.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 + 9 (- λ - 241) - λ (- λ - 241) - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.4.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 + 9 (- λ) + 9 \cdot - 241 - λ (- λ - 241) - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.4.3

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 + 9 (- λ) + 9 \cdot - 241 - λ (- λ) - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5

Sederhanakan dan gabungkan suku-suku sejenis.

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Langkah 1.5.3.5.1.5.1

Sederhanakan setiap suku.

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Langkah 1.5.3.5.1.5.1.1

Kalikan $- 1$ dengan $9$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ + 9 \cdot - 241 - λ (- λ) - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.2

Kalikan $9$ dengan $- 241$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 - λ (- λ) - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.3

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 - 1 \cdot - 1 λ \cdot λ - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.4

Kalikan $λ$ dengan $λ$ dengan menambahkan eksponennya.

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Langkah 1.5.3.5.1.5.1.4.1

Pindahkan $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 - 1 \cdot - 1 (λ \cdot λ) - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.4.2

Kalikan $λ$ dengan $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 - 1 \cdot - 1 λ^{2} - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.5

Kalikan $- 1$ dengan $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 + 1 λ^{2} - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.6

Kalikan $λ^{2}$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 + λ^{2} - λ \cdot - 241 - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.1.7

Kalikan $- 241$ dengan $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 - 9 λ - 2169 + λ^{2} + 241 λ - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.5.2

Tambahkan $- 9 λ$ dan $241 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 + 232 λ - 2169 + λ^{2} - 3 \cdot - 1) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.1.6

Kalikan $- 3$ dengan $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (32 λ + 6352 + 232 λ - 2169 + λ^{2} + 3) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.2

Tambahkan $32 λ$ dan $232 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (264 λ + 6352 - 2169 + λ^{2} + 3) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.3

Kurangi $2169$ dengan $6352$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (264 λ + 4183 + λ^{2} + 3) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.4

Tambahkan $4183$ dan $3$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (264 λ + λ^{2} + 4186) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.3.5.5

Susun kembali $264 λ$ dan $λ^{2}$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 | \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} | - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4

Evaluasi $| \begin{array}{c} 1 & 56 - λ & 85 \\ 16 & 7 & 3 \\ 4 & 5 & 99 - λ \end{array} |$ .

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Langkah 1.5.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Langkah 1.5.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Langkah 1.5.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Langkah 1.5.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} |$

Langkah 1.5.4.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} |$

Langkah 1.5.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} |$

Langkah 1.5.4.1.6

Multiply element $a_{12}$ by its cofactor.

$- (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} |$

Langkah 1.5.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |$

Langkah 1.5.4.1.8

Multiply element $a_{13}$ by its cofactor.

$85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |$

Langkah 1.5.4.1.9

Add the terms together.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 | \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} | - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.2

Evaluasi $| \begin{array}{c} 7 & 3 \\ 5 & 99 - λ \end{array} |$ .

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Langkah 1.5.4.2.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (7 (99 - λ) - 5 \cdot 3) - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.2.2

Sederhanakan determinannya.

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Langkah 1.5.4.2.2.1

Sederhanakan setiap suku.

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Langkah 1.5.4.2.2.1.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (7 \cdot 99 + 7 (- λ) - 5 \cdot 3) - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.2.2.1.2

Kalikan $7$ dengan $99$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (693 + 7 (- λ) - 5 \cdot 3) - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.2.2.1.3

Kalikan $- 1$ dengan $7$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (693 - 7 λ - 5 \cdot 3) - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.2.2.1.4

Kalikan $- 5$ dengan $3$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (693 - 7 λ - 15) - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.2.2.2

Kurangi $15$ dengan $693$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) | \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} | + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.3

Evaluasi $| \begin{array}{c} 16 & 3 \\ 4 & 99 - λ \end{array} |$ .

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Langkah 1.5.4.3.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (16 (99 - λ) - 4 \cdot 3) + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.3.2

Sederhanakan determinannya.

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Langkah 1.5.4.3.2.1

Sederhanakan setiap suku.

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Langkah 1.5.4.3.2.1.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (16 \cdot 99 + 16 (- λ) - 4 \cdot 3) + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.3.2.1.2

Kalikan $16$ dengan $99$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (1584 + 16 (- λ) - 4 \cdot 3) + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.3.2.1.3

Kalikan $- 1$ dengan $16$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (1584 - 16 λ - 4 \cdot 3) + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.3.2.1.4

Kalikan $- 4$ dengan $3$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (1584 - 16 λ - 12) + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.3.2.2

Kurangi $12$ dengan $1584$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (- 16 λ + 1572) + 85 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.4

Evaluasi $| \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |$ .

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Langkah 1.5.4.4.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (- 16 λ + 1572) + 85 (16 \cdot 5 - 4 \cdot 7)) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.4.2

Sederhanakan determinannya.

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Langkah 1.5.4.4.2.1

Sederhanakan setiap suku.

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Langkah 1.5.4.4.2.1.1

Kalikan $16$ dengan $5$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (- 16 λ + 1572) + 85 (80 - 4 \cdot 7)) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.4.2.1.2

Kalikan $- 4$ dengan $7$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (- 16 λ + 1572) + 85 (80 - 28)) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.4.2.2

Kurangi $28$ dengan $80$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (1 (- 7 λ + 678) - (56 - λ) (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5

Sederhanakan determinannya.

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Langkah 1.5.4.5.1

Sederhanakan setiap suku.

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Langkah 1.5.4.5.1.1

Kalikan $- 7 λ + 678$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 - (56 - λ) (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + (- 1 \cdot 56 - - λ) (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.3

Kalikan $- 1$ dengan $56$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + (- 56 - - λ) (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.4

Kalikan $- - λ$ .

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Langkah 1.5.4.5.1.4.1

Kalikan $- 1$ dengan $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + (- 56 + 1 λ) (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.4.2

Kalikan $λ$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + (- 56 + λ) (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.5

Perluas $(- 56 + λ) (- 16 λ + 1572)$ menggunakan Metode FOIL.

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Langkah 1.5.4.5.1.5.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 - 56 (- 16 λ + 1572) + λ (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.5.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 - 56 (- 16 λ) - 56 \cdot 1572 + λ (- 16 λ + 1572) + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.5.3

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 - 56 (- 16 λ) - 56 \cdot 1572 + λ (- 16 λ) + λ \cdot 1572 + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6

Sederhanakan dan gabungkan suku-suku sejenis.

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Langkah 1.5.4.5.1.6.1

Sederhanakan setiap suku.

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Langkah 1.5.4.5.1.6.1.1

Kalikan $- 16$ dengan $- 56$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 896 λ - 56 \cdot 1572 + λ (- 16 λ) + λ \cdot 1572 + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6.1.2

Kalikan $- 56$ dengan $1572$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 896 λ - 88032 + λ (- 16 λ) + λ \cdot 1572 + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6.1.3

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 896 λ - 88032 - 16 λ \cdot λ + λ \cdot 1572 + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6.1.4

Kalikan $λ$ dengan $λ$ dengan menambahkan eksponennya.

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Langkah 1.5.4.5.1.6.1.4.1

Pindahkan $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 896 λ - 88032 - 16 (λ \cdot λ) + λ \cdot 1572 + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6.1.4.2

Kalikan $λ$ dengan $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 896 λ - 88032 - 16 λ^{2} + λ \cdot 1572 + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6.1.5

Pindahkan $1572$ ke sebelah kiri $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 896 λ - 88032 - 16 λ^{2} + 1572 λ + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.6.2

Tambahkan $896 λ$ dan $1572 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 2468 λ - 88032 - 16 λ^{2} + 85 \cdot 52) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.1.7

Kalikan $85$ dengan $52$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 7 λ + 678 + 2468 λ - 88032 - 16 λ^{2} + 4420) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.2

Tambahkan $- 7 λ$ dan $2468 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (2461 λ + 678 - 88032 - 16 λ^{2} + 4420) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.3

Kurangi $88032$ dengan $678$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (2461 λ - 87354 - 16 λ^{2} + 4420) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.4

Tambahkan $- 87354$ dan $4420$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (2461 λ - 16 λ^{2} - 82934) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.4.5.5

Susun kembali $2461 λ$ dan $- 16 λ^{2}$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 | \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$

Langkah 1.5.5

Evaluasi $| \begin{array}{c} 1 & 56 - λ & 2 \\ 16 & 7 & 9 - λ \\ 4 & 5 & 7 \end{array} |$ .

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Langkah 1.5.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Langkah 1.5.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Langkah 1.5.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Langkah 1.5.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |$

Langkah 1.5.5.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |$

Langkah 1.5.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} |$

Langkah 1.5.5.1.6

Multiply element $a_{12}$ by its cofactor.

$- (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} |$

Langkah 1.5.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |$

Langkah 1.5.5.1.8

Multiply element $a_{13}$ by its cofactor.

$2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |$

Langkah 1.5.5.1.9

Add the terms together.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 | \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} | - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2

Evaluasi $| \begin{array}{c} 7 & 9 - λ \\ 5 & 7 \end{array} |$ .

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Langkah 1.5.5.2.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (7 \cdot 7 - 5 (9 - λ)) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2.2

Sederhanakan determinannya.

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Langkah 1.5.5.2.2.1

Sederhanakan setiap suku.

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Langkah 1.5.5.2.2.1.1

Kalikan $7$ dengan $7$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (49 - 5 (9 - λ)) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2.2.1.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (49 - 5 \cdot 9 - 5 (- λ)) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2.2.1.3

Kalikan $- 5$ dengan $9$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (49 - 45 - 5 (- λ)) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2.2.1.4

Kalikan $- 1$ dengan $- 5$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (49 - 45 + 5 λ) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2.2.2

Kurangi $45$ dengan $49$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (4 + 5 λ) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.2.2.3

Susun kembali $4$ dan $5 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) | \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} | + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3

Evaluasi $| \begin{array}{c} 16 & 9 - λ \\ 4 & 7 \end{array} |$ .

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Langkah 1.5.5.3.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (16 \cdot 7 - 4 (9 - λ)) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3.2

Sederhanakan determinannya.

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Langkah 1.5.5.3.2.1

Sederhanakan setiap suku.

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Langkah 1.5.5.3.2.1.1

Kalikan $16$ dengan $7$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (112 - 4 (9 - λ)) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3.2.1.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (112 - 4 \cdot 9 - 4 (- λ)) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3.2.1.3

Kalikan $- 4$ dengan $9$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (112 - 36 - 4 (- λ)) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3.2.1.4

Kalikan $- 1$ dengan $- 4$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (112 - 36 + 4 λ) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3.2.2

Kurangi $36$ dengan $112$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (76 + 4 λ) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.3.2.3

Susun kembali $76$ dan $4 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (4 λ + 76) + 2 | \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |)$

Langkah 1.5.5.4

Evaluasi $| \begin{array}{c} 16 & 7 \\ 4 & 5 \end{array} |$ .

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Langkah 1.5.5.4.1

Determinan dari matriks $2 \times 2$ dapat dicari menggunakan rumus $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (4 λ + 76) + 2 (16 \cdot 5 - 4 \cdot 7))$

Langkah 1.5.5.4.2

Sederhanakan determinannya.

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Langkah 1.5.5.4.2.1

Sederhanakan setiap suku.

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Langkah 1.5.5.4.2.1.1

Kalikan $16$ dengan $5$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (4 λ + 76) + 2 (80 - 4 \cdot 7))$

Langkah 1.5.5.4.2.1.2

Kalikan $- 4$ dengan $7$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (4 λ + 76) + 2 (80 - 28))$

Langkah 1.5.5.4.2.2

Kurangi $28$ dengan $80$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (1 (5 λ + 4) - (56 - λ) (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5

Sederhanakan determinannya.

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Langkah 1.5.5.5.1

Sederhanakan setiap suku.

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Langkah 1.5.5.5.1.1

Kalikan $5 λ + 4$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - (56 - λ) (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 + (- 1 \cdot 56 - - λ) (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.3

Kalikan $- 1$ dengan $56$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 + (- 56 - - λ) (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.4

Kalikan $- - λ$ .

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Langkah 1.5.5.5.1.4.1

Kalikan $- 1$ dengan $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 + (- 56 + 1 λ) (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.4.2

Kalikan $λ$ dengan $1$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 + (- 56 + λ) (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.5

Perluas $(- 56 + λ) (4 λ + 76)$ menggunakan Metode FOIL.

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Langkah 1.5.5.5.1.5.1

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 56 (4 λ + 76) + λ (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.5.2

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 56 (4 λ) - 56 \cdot 76 + λ (4 λ + 76) + 2 \cdot 52)$

Langkah 1.5.5.5.1.5.3

Terapkan sifat distributif.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 56 (4 λ) - 56 \cdot 76 + λ (4 λ) + λ \cdot 76 + 2 \cdot 52)$

Langkah 1.5.5.5.1.6

Sederhanakan dan gabungkan suku-suku sejenis.

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Langkah 1.5.5.5.1.6.1

Sederhanakan setiap suku.

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Langkah 1.5.5.5.1.6.1.1

Kalikan $4$ dengan $- 56$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 224 λ - 56 \cdot 76 + λ (4 λ) + λ \cdot 76 + 2 \cdot 52)$

Langkah 1.5.5.5.1.6.1.2

Kalikan $- 56$ dengan $76$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 224 λ - 4256 + λ (4 λ) + λ \cdot 76 + 2 \cdot 52)$

Langkah 1.5.5.5.1.6.1.3

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 224 λ - 4256 + 4 λ \cdot λ + λ \cdot 76 + 2 \cdot 52)$

Langkah 1.5.5.5.1.6.1.4

Kalikan $λ$ dengan $λ$ dengan menambahkan eksponennya.

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Langkah 1.5.5.5.1.6.1.4.1

Pindahkan $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 224 λ - 4256 + 4 (λ \cdot λ) + λ \cdot 76 + 2 \cdot 52)$

Langkah 1.5.5.5.1.6.1.4.2

Kalikan $λ$ dengan $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 224 λ - 4256 + 4 λ^{2} + λ \cdot 76 + 2 \cdot 52)$

Langkah 1.5.5.5.1.6.1.5

Pindahkan $76$ ke sebelah kiri $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 224 λ - 4256 + 4 λ^{2} + 76 λ + 2 \cdot 52)$

Langkah 1.5.5.5.1.6.2

Tambahkan $- 224 λ$ dan $76 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 148 λ - 4256 + 4 λ^{2} + 2 \cdot 52)$

Langkah 1.5.5.5.1.7

Kalikan $2$ dengan $52$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (5 λ + 4 - 148 λ - 4256 + 4 λ^{2} + 104)$

Langkah 1.5.5.5.2

Kurangi $148 λ$ dengan $5 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (- 143 λ + 4 - 4256 + 4 λ^{2} + 104)$

Langkah 1.5.5.5.3

Kurangi $4256$ dengan $4$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (- 143 λ - 4252 + 4 λ^{2} + 104)$

Langkah 1.5.5.5.4

Tambahkan $- 4252$ dan $104$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (- 143 λ + 4 λ^{2} - 4148)$

Langkah 1.5.5.5.5

Susun kembali $- 143 λ$ dan $4 λ^{2}$ .

$p (λ) = (1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704) - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6

Sederhanakan determinannya.

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Langkah 1.5.6.1

Sederhanakan setiap suku.

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Langkah 1.5.6.1.1

Perluas $(1 - λ) (- λ^{3} + 164 λ^{2} - 6479 λ + 47704)$ dengan mengalikan setiap suku dalam pernyataan pertama dengan setiap suku dalam pernyataan kedua.

$p (λ) = 1 (- λ^{3}) + 1 (164 λ^{2}) + 1 (- 6479 λ) + 1 \cdot 47704 - λ (- λ^{3}) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2

Sederhanakan setiap suku.

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Langkah 1.5.6.1.2.1

Kalikan $- λ^{3}$ dengan $1$ .

$p (λ) = - λ^{3} + 1 (164 λ^{2}) + 1 (- 6479 λ) + 1 \cdot 47704 - λ (- λ^{3}) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.2

Kalikan $164 λ^{2}$ dengan $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} + 1 (- 6479 λ) + 1 \cdot 47704 - λ (- λ^{3}) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.3

Kalikan $- 6479 λ$ dengan $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 1 \cdot 47704 - λ (- λ^{3}) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.4

Kalikan $47704$ dengan $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 - λ (- λ^{3}) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.5

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 - 1 \cdot - 1 λ \cdot λ^{3} - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.6

Kalikan $λ$ dengan $λ^{3}$ dengan menambahkan eksponennya.

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Langkah 1.5.6.1.2.6.1

Pindahkan $λ^{3}$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 - 1 \cdot - 1 (λ^{3} λ) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.6.2

Kalikan $λ^{3}$ dengan $λ$ .

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Langkah 1.5.6.1.2.6.2.1

Naikkan $λ$ menjadi pangkat $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 - 1 \cdot - 1 (λ^{3} λ^{1}) - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.6.2.2

Gunakan kaidah pangkat $a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 - 1 \cdot - 1 λ^{3 + 1} - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.6.3

Tambahkan $3$ dan $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 - 1 \cdot - 1 λ^{4} - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.7

Kalikan $- 1$ dengan $- 1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + 1 λ^{4} - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.8

Kalikan $λ^{4}$ dengan $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - λ (164 λ^{2}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.9

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 1 \cdot 164 λ \cdot λ^{2} - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.10

Kalikan $λ$ dengan $λ^{2}$ dengan menambahkan eksponennya.

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Langkah 1.5.6.1.2.10.1

Pindahkan $λ^{2}$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 1 \cdot 164 (λ^{2} λ) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.10.2

Kalikan $λ^{2}$ dengan $λ$ .

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Langkah 1.5.6.1.2.10.2.1

Naikkan $λ$ menjadi pangkat $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 1 \cdot 164 (λ^{2} λ^{1}) - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.10.2.2

Gunakan kaidah pangkat $a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 1 \cdot 164 λ^{2 + 1} - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.10.3

Tambahkan $2$ dan $1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 1 \cdot 164 λ^{3} - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.11

Kalikan $- 1$ dengan $164$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 164 λ^{3} - λ (- 6479 λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.12

Tulis kembali menggunakan sifat komutatif dari perkalian.

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 164 λ^{3} - 1 \cdot - 6479 λ \cdot λ - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.13

Kalikan $λ$ dengan $λ$ dengan menambahkan eksponennya.

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Langkah 1.5.6.1.2.13.1

Pindahkan $λ$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 164 λ^{3} - 1 \cdot - 6479 (λ \cdot λ) - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.13.2

Kalikan $λ$ dengan $λ$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 164 λ^{3} - 1 \cdot - 6479 λ^{2} - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.14

Kalikan $- 1$ dengan $- 6479$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 164 λ^{3} + 6479 λ^{2} - λ \cdot 47704 - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.2.15

Kalikan $47704$ dengan $- 1$ .

$p (λ) = - λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} - 164 λ^{3} + 6479 λ^{2} - 47704 λ - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.3

Kurangi $164 λ^{3}$ dengan $- λ^{3}$ .

$p (λ) = - 165 λ^{3} + 164 λ^{2} - 6479 λ + 47704 + λ^{4} + 6479 λ^{2} - 47704 λ - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.4

Tambahkan $164 λ^{2}$ dan $6479 λ^{2}$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 6479 λ + 47704 + λ^{4} - 47704 λ - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.5

Kurangi $47704 λ$ dengan $- 6479 λ$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 (λ^{2} + 264 λ + 4186) + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.6

Terapkan sifat distributif.

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 2 (264 λ) - 2 \cdot 4186 + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.7

Sederhanakan.

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Langkah 1.5.6.1.7.1

Kalikan $264$ dengan $- 2$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 2 \cdot 4186 + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.7.2

Kalikan $- 2$ dengan $4186$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 + 6 (- 16 λ^{2} + 2461 λ - 82934) - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.8

Terapkan sifat distributif.

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 + 6 (- 16 λ^{2}) + 6 (2461 λ) + 6 \cdot - 82934 - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.9

Sederhanakan.

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Langkah 1.5.6.1.9.1

Kalikan $- 16$ dengan $6$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 6 (2461 λ) + 6 \cdot - 82934 - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.9.2

Kalikan $2461$ dengan $6$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 14766 λ + 6 \cdot - 82934 - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.9.3

Kalikan $6$ dengan $- 82934$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 14766 λ - 497604 - 8 (4 λ^{2} - 143 λ - 4148)$

Langkah 1.5.6.1.10

Terapkan sifat distributif.

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 14766 λ - 497604 - 8 (4 λ^{2}) - 8 (- 143 λ) - 8 \cdot - 4148$

Langkah 1.5.6.1.11

Sederhanakan.

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Langkah 1.5.6.1.11.1

Kalikan $4$ dengan $- 8$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 14766 λ - 497604 - 32 λ^{2} - 8 (- 143 λ) - 8 \cdot - 4148$

Langkah 1.5.6.1.11.2

Kalikan $- 143$ dengan $- 8$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 14766 λ - 497604 - 32 λ^{2} + 1144 λ - 8 \cdot - 4148$

Langkah 1.5.6.1.11.3

Kalikan $- 8$ dengan $- 4148$ .

$p (λ) = - 165 λ^{3} + 6643 λ^{2} - 54183 λ + 47704 + λ^{4} - 2 λ^{2} - 528 λ - 8372 - 96 λ^{2} + 14766 λ - 497604 - 32 λ^{2} + 1144 λ + 33184$

Langkah 1.5.6.2

Kurangi $2 λ^{2}$ dengan $6643 λ^{2}$ .

$p (λ) = - 165 λ^{3} + 6641 λ^{2} - 54183 λ + 47704 + λ^{4} - 528 λ - 8372 - 96 λ^{2} + 14766 λ - 497604 - 32 λ^{2} + 1144 λ + 33184$

Langkah 1.5.6.3

Kurangi $96 λ^{2}$ dengan $6641 λ^{2}$ .

$p (λ) = - 165 λ^{3} + 6545 λ^{2} - 54183 λ + 47704 + λ^{4} - 528 λ - 8372 + 14766 λ - 497604 - 32 λ^{2} + 1144 λ + 33184$

Langkah 1.5.6.4

Kurangi $32 λ^{2}$ dengan $6545 λ^{2}$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 54183 λ + 47704 + λ^{4} - 528 λ - 8372 + 14766 λ - 497604 + 1144 λ + 33184$

Langkah 1.5.6.5

Kurangi $528 λ$ dengan $- 54183 λ$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 54711 λ + 47704 + λ^{4} - 8372 + 14766 λ - 497604 + 1144 λ + 33184$

Langkah 1.5.6.6

Tambahkan $- 54711 λ$ dan $14766 λ$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 39945 λ + 47704 + λ^{4} - 8372 - 497604 + 1144 λ + 33184$

Langkah 1.5.6.7

Tambahkan $- 39945 λ$ dan $1144 λ$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 38801 λ + 47704 + λ^{4} - 8372 - 497604 + 33184$

Langkah 1.5.6.8

Kurangi $8372$ dengan $47704$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 38801 λ + λ^{4} + 39332 - 497604 + 33184$

Langkah 1.5.6.9

Kurangi $497604$ dengan $39332$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 38801 λ + λ^{4} - 458272 + 33184$

Langkah 1.5.6.10

Tambahkan $- 458272$ dan $33184$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} - 38801 λ + λ^{4} - 425088$

Langkah 1.5.6.11

Pindahkan $- 38801 λ$ .

$p (λ) = - 165 λ^{3} + 6513 λ^{2} + λ^{4} - 38801 λ - 425088$

Langkah 1.5.6.12

Pindahkan $6513 λ^{2}$ .

$p (λ) = - 165 λ^{3} + λ^{4} + 6513 λ^{2} - 38801 λ - 425088$

Langkah 1.5.6.13

Susun kembali $- 165 λ^{3}$ dan $λ^{4}$ .

$p (λ) = λ^{4} - 165 λ^{3} + 6513 λ^{2} - 38801 λ - 425088$

Langkah 1.6

Atur polinomial karakteristiknya agar sama dengan $0$ untuk menemukan nilai eigen $λ$ .

$λ^{4} - 165 λ^{3} + 6513 λ^{2} - 38801 λ - 425088 = 0$

Langkah 1.7

Selesaikan $λ$ .

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Langkah 1.7.1

Gambarkan setiap sisi persamaan. Penyelesaiannya adalah nilai x dari titik perpotongan.

$λ \approx - 5.39042551, 15.81232455, 45.88268919$

Langkah 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where $N$ is the null space and $I$ is the identity matrix.

$ε_{A} = N (A - λ I_{4})$

Langkah 3

Find the eigenvector using the eigenvalue $λ = - 5.39042551$ .

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Langkah 3.1

Substitusikan nilai-nilai yang diketahui ke dalam rumusnya.

$N ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + 5.39042551 [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Langkah 3.2

Sederhanakan.

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Langkah 3.2.1

Sederhanakan setiap suku.

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Langkah 3.2.1.1

Kalikan $5.39042551$ dengan setiap elemen di dalam matriks tersebut.

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2

Sederhanakan setiap elemen dalam matriks.

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Langkah 3.2.1.2.1

Kalikan $5.39042551$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.2

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.3

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.4

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.5

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.6

Kalikan $5.39042551$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.7

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.8

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.9

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.10

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 0 & 5.39042551 \cdot 1 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.11

Kalikan $5.39042551$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 0 & 5.39042551 & 5.39042551 \cdot 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.12

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 0 & 5.39042551 & 0 \\ 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.13

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 0 & 5.39042551 & 0 \\ 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.14

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 0 & 5.39042551 & 0 \\ 0 & 0 & 5.39042551 \cdot 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.15

Kalikan $5.39042551$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} 5.39042551 & 0 & 0 & 0 \\ 0 & 5.39042551 & 0 & 0 \\ 0 & 0 & 5.39042551 & 0 \\ 0 & 0 & 0 & 5.39042551 \cdot 1 \end{array}]$

Langkah 3.2.1.2.16

Kalikan $5.39042551$ dengan $1$ .

Langkah 3.2.2

Tambahkan elemen yang seletak.

$[\begin{array}{c} 1 + 5.39042551 & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 + 5.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3

Simplify each element.

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Langkah 3.2.3.1

Tambahkan $1$ dan $5.39042551$ .

$[\begin{array}{c} 6.39042551 & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 + 5.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.2

Tambahkan $2$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 + 5.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.3

Tambahkan $6$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 + 0 \\ 1 + 0 & 56 + 5.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.4

Tambahkan $8$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 + 0 & 56 + 5.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.5

Tambahkan $1$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 56 + 5.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.6

Tambahkan $56$ dan $5.39042551$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.7

Tambahkan $2$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.8

Tambahkan $85$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 + 0 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.9

Tambahkan $16$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 + 0 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.10

Tambahkan $7$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 9 + 5.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.11

Tambahkan $9$ dan $5.39042551$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 14.39042551 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.12

Tambahkan $3$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 14.39042551 & 3 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.13

Tambahkan $4$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 14.39042551 & 3 \\ 4 & 5 + 0 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.14

Tambahkan $5$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 14.39042551 & 3 \\ 4 & 5 & 7 + 0 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.15

Tambahkan $7$ dan $0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 14.39042551 & 3 \\ 4 & 5 & 7 & 99 + 5.39042551 \end{array}]$

Langkah 3.2.3.16

Tambahkan $99$ dan $5.39042551$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 \\ 1 & 61.39042551 & 2 & 85 \\ 16 & 7 & 14.39042551 & 3 \\ 4 & 5 & 7 & 104.39042551 \end{array}]$

Langkah 3.3

Find the null space when $λ = - 5.39042551$ .

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Langkah 3.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 6.39042551 & 2 & 6 & 8 & 0 \\ 1 & 61.39042551 & 2 & 85 & 0 \\ 16 & 7 & 14.39042551 & 3 & 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 3.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{6.39042551}$ to make the entry at $1, 1$ a $1$ .

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Langkah 3.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6.39042551}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6.39042551}{6.39042551} & \frac{2}{6.39042551} & \frac{6}{6.39042551} & \frac{8}{6.39042551} & \frac{0}{6.39042551} \\ 1 & 61.39042551 & 2 & 85 & 0 \\ 16 & 7 & 14.39042551 & 3 & 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 1 & 61.39042551 & 2 & 85 & 0 \\ 16 & 7 & 14.39042551 & 3 & 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 3.3.2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 1 - 1 & 61.39042551 - 0.3129682 & 2 - 0.93890461 & 85 - 1.25187281 & 0 - 0 \\ 16 & 7 & 14.39042551 & 3 & 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 61.0774573 & 1.06109538 & 83.74812718 & 0 \\ 16 & 7 & 14.39042551 & 3 & 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2.3

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 3.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 61.0774573 & 1.06109538 & 83.74812718 & 0 \\ 16 - 16 \cdot 1 & 7 - 16 \cdot 0.3129682 & 14.39042551 - 16 \cdot 0.93890461 & 3 - 16 \cdot 1.25187281 & 0 - 16 \cdot 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 61.0774573 & 1.06109538 & 83.74812718 & 0 \\ 0 & 1.99250872 & - 0.63204831 & - 17.0299651 & 0 \\ 4 & 5 & 7 & 104.39042551 & 0 \end{array}]$

Langkah 3.3.2.4

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Langkah 3.3.2.4.1

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 61.0774573 & 1.06109538 & 83.74812718 & 0 \\ 0 & 1.99250872 & - 0.63204831 & - 17.0299651 & 0 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 0.3129682 & 7 - 4 \cdot 0.93890461 & 104.39042551 - 4 \cdot 1.25187281 & 0 - 4 \cdot 0 \end{array}]$

Langkah 3.3.2.4.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 61.0774573 & 1.06109538 & 83.74812718 & 0 \\ 0 & 1.99250872 & - 0.63204831 & - 17.0299651 & 0 \\ 0 & 3.74812718 & 3.24438154 & 99.38293423 & 0 \end{array}]$

Langkah 3.3.2.5

Multiply each element of $R_{2}$ by $\frac{1}{61.0774573}$ to make the entry at $2, 2$ a $1$ .

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Langkah 3.3.2.5.1

Multiply each element of $R_{2}$ by $\frac{1}{61.0774573}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ \frac{0}{61.0774573} & \frac{61.0774573}{61.0774573} & \frac{1.06109538}{61.0774573} & \frac{83.74812718}{61.0774573} & \frac{0}{61.0774573} \\ 0 & 1.99250872 & - 0.63204831 & - 17.0299651 & 0 \\ 0 & 3.74812718 & 3.24438154 & 99.38293423 & 0 \end{array}]$

Langkah 3.3.2.5.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 1.99250872 & - 0.63204831 & - 17.0299651 & 0 \\ 0 & 3.74812718 & 3.24438154 & 99.38293423 & 0 \end{array}]$

Langkah 3.3.2.6

Perform the row operation $R_{3} = R_{3} - 1.99250872 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 3.3.2.6.1

Perform the row operation $R_{3} = R_{3} - 1.99250872 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 - 1.99250872 \cdot 0 & 1.99250872 - 1.99250872 \cdot 1 & - 0.63204831 - 1.99250872 \cdot 0.01737294 & - 17.0299651 - 1.99250872 \cdot 1.371179 & 0 - 1.99250872 \cdot 0 \\ 0 & 3.74812718 & 3.24438154 & 99.38293423 & 0 \end{array}]$

Langkah 3.3.2.6.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & - 0.66666406 & - 19.76205123 & 0 \\ 0 & 3.74812718 & 3.24438154 & 99.38293423 & 0 \end{array}]$

Langkah 3.3.2.7

Perform the row operation $R_{4} = R_{4} - 3.74812718 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Langkah 3.3.2.7.1

Perform the row operation $R_{4} = R_{4} - 3.74812718 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & - 0.66666406 & - 19.76205123 & 0 \\ 0 - 3.74812718 \cdot 0 & 3.74812718 - 3.74812718 \cdot 1 & 3.24438154 - 3.74812718 \cdot 0.01737294 & 99.38293423 - 3.74812718 \cdot 1.371179 & 0 - 3.74812718 \cdot 0 \end{array}]$

Langkah 3.3.2.7.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & - 0.66666406 & - 19.76205123 & 0 \\ 0 & 0 & 3.17926553 & 94.24358093 & 0 \end{array}]$

Langkah 3.3.2.8

Multiply each element of $R_{3}$ by $\frac{1}{- 0.66666406}$ to make the entry at $3, 3$ a $1$ .

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Langkah 3.3.2.8.1

Multiply each element of $R_{3}$ by $\frac{1}{- 0.66666406}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ \frac{0}{- 0.66666406} & \frac{0}{- 0.66666406} & \frac{- 0.66666406}{- 0.66666406} & \frac{- 19.76205123}{- 0.66666406} & \frac{0}{- 0.66666406} \\ 0 & 0 & 3.17926553 & 94.24358093 & 0 \end{array}]$

Langkah 3.3.2.8.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & 1 & 29.64319274 & 0 \\ 0 & 0 & 3.17926553 & 94.24358093 & 0 \end{array}]$

Langkah 3.3.2.9

Perform the row operation $R_{4} = R_{4} - 3.17926553 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Langkah 3.3.2.9.1

Perform the row operation $R_{4} = R_{4} - 3.17926553 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & 1 & 29.64319274 & 0 \\ 0 - 3.17926553 \cdot 0 & 0 - 3.17926553 \cdot 0 & 3.17926553 - 3.17926553 \cdot 1 & 94.24358093 - 3.17926553 \cdot 29.64319274 & 0 - 3.17926553 \cdot 0 \end{array}]$

Langkah 3.3.2.9.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & 1 & 29.64319274 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Langkah 3.3.2.10

Multiply each element of $R_{4}$ by $\frac{1}{6.105265 \cdot 10^{- 13}}$ to make the entry at $4, 4$ a $1$ .

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Langkah 3.3.2.10.1

Multiply each element of $R_{4}$ by $\frac{1}{6.105265 \cdot 10^{- 13}}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & 1 & 29.64319274 & 0 \\ \frac{0}{6.105265 \cdot 10^{- 13}} & \frac{0}{6.105265 \cdot 10^{- 13}} & \frac{0}{6.105265 \cdot 10^{- 13}} & \frac{0}{6.105265 \cdot 10^{- 13}} & \frac{0}{6.105265 \cdot 10^{- 13}} \end{array}]$

Langkah 3.3.2.10.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & 1 & 29.64319274 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.11

Perform the row operation $R_{3} = R_{3} - 29.64319274 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Langkah 3.3.2.11.1

Perform the row operation $R_{3} = R_{3} - 29.64319274 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 - 29.64319274 \cdot 0 & 0 - 29.64319274 \cdot 0 & 1 - 29.64319274 \cdot 0 & 29.64319274 - 29.64319274 \cdot 1 & 0 - 29.64319274 \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.11.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 1.371179 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.12

Perform the row operation $R_{2} = R_{2} - 1.371179 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Langkah 3.3.2.12.1

Perform the row operation $R_{2} = R_{2} - 1.371179 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 - 1.371179 \cdot 0 & 1 - 1.371179 \cdot 0 & 0.01737294 - 1.371179 \cdot 0 & 1.371179 - 1.371179 \cdot 1 & 0 - 1.371179 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.12.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 1.25187281 & 0 \\ 0 & 1 & 0.01737294 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.13

Perform the row operation $R_{1} = R_{1} - 1.25187281 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Langkah 3.3.2.13.1

Perform the row operation $R_{1} = R_{1} - 1.25187281 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 1.25187281 \cdot 0 & 0.3129682 - 1.25187281 \cdot 0 & 0.93890461 - 1.25187281 \cdot 0 & 1.25187281 - 1.25187281 \cdot 1 & 0 - 1.25187281 \cdot 0 \\ 0 & 1 & 0.01737294 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.13.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 0 & 0 \\ 0 & 1 & 0.01737294 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.14

Perform the row operation $R_{2} = R_{2} - 0.01737294 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Langkah 3.3.2.14.1

Perform the row operation $R_{2} = R_{2} - 0.01737294 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 0 & 0 \\ 0 - 0.01737294 \cdot 0 & 1 - 0.01737294 \cdot 0 & 0.01737294 - 0.01737294 \cdot 1 & 0 - 0.01737294 \cdot 0 & 0 - 0.01737294 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.14.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0.93890461 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.15

Perform the row operation $R_{1} = R_{1} - 0.93890461 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Langkah 3.3.2.15.1

Perform the row operation $R_{1} = R_{1} - 0.93890461 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.93890461 \cdot 0 & 0.3129682 - 0.93890461 \cdot 0 & 0.93890461 - 0.93890461 \cdot 1 & 0 - 0.93890461 \cdot 0 & 0 - 0.93890461 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.15.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0.3129682 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.16

Perform the row operation $R_{1} = R_{1} - 0.3129682 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 3.3.2.16.1

Perform the row operation $R_{1} = R_{1} - 0.3129682 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.3129682 \cdot 0 & 0.3129682 - 0.3129682 \cdot 1 & 0 - 0.3129682 \cdot 0 & 0 - 0.3129682 \cdot 0 & 0 - 0.3129682 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.2.16.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 3.3.3

Use the result matrix to declare the final solution to the system of equations.

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

Langkah 3.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]$

Langkah 3.3.5

Write as a solution set.

${[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]}$

Langkah 4

Find the eigenvector using the eigenvalue $λ = 15.81232455$ .

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Langkah 4.1

Substitusikan nilai-nilai yang diketahui ke dalam rumusnya.

$N ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] - 15.81232455 [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Langkah 4.2

Sederhanakan.

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Langkah 4.2.1

Sederhanakan setiap suku.

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Langkah 4.2.1.1

Kalikan $- 15.81232455$ dengan setiap elemen di dalam matriks tersebut.

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2

Sederhanakan setiap elemen dalam matriks.

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Langkah 4.2.1.2.1

Kalikan $- 15.81232455$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.2

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.3

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.4

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.5

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.6

Kalikan $- 15.81232455$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.7

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.8

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.9

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.10

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ 0 & 0 & - 15.81232455 \cdot 1 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.11

Kalikan $- 15.81232455$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ 0 & 0 & - 15.81232455 & - 15.81232455 \cdot 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.12

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ 0 & 0 & - 15.81232455 & 0 \\ - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.13

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ 0 & 0 & - 15.81232455 & 0 \\ 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.14

Kalikan $- 15.81232455$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 15.81232455 & 0 & 0 & 0 \\ 0 & - 15.81232455 & 0 & 0 \\ 0 & 0 & - 15.81232455 & 0 \\ 0 & 0 & - 15.81232455 \cdot 0 & - 15.81232455 \cdot 1 \end{array}]$

Langkah 4.2.1.2.15

Kalikan $- 15.81232455$ dengan $0$ .

Langkah 4.2.1.2.16

Kalikan $- 15.81232455$ dengan $1$ .

Langkah 4.2.2

Tambahkan elemen yang seletak.

$[\begin{array}{c} 1 - 15.81232455 & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - 15.81232455 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3

Simplify each element.

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Langkah 4.2.3.1

Kurangi $15.81232455$ dengan $1$ .

$[\begin{array}{c} - 14.81232455 & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - 15.81232455 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.2

Tambahkan $2$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - 15.81232455 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.3

Tambahkan $6$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 + 0 \\ 1 + 0 & 56 - 15.81232455 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.4

Tambahkan $8$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 + 0 & 56 - 15.81232455 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.5

Tambahkan $1$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 56 - 15.81232455 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.6

Kurangi $15.81232455$ dengan $56$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.7

Tambahkan $2$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.8

Tambahkan $85$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 + 0 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.9

Tambahkan $16$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 + 0 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.10

Tambahkan $7$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & 9 - 15.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.11

Kurangi $15.81232455$ dengan $9$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & - 6.81232455 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.12

Tambahkan $3$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & - 6.81232455 & 3 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.13

Tambahkan $4$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & - 6.81232455 & 3 \\ 4 & 5 + 0 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.14

Tambahkan $5$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & - 6.81232455 & 3 \\ 4 & 5 & 7 + 0 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.15

Tambahkan $7$ dan $0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & - 6.81232455 & 3 \\ 4 & 5 & 7 & 99 - 15.81232455 \end{array}]$

Langkah 4.2.3.16

Kurangi $15.81232455$ dengan $99$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 \\ 1 & 40.18767544 & 2 & 85 \\ 16 & 7 & - 6.81232455 & 3 \\ 4 & 5 & 7 & 83.18767544 \end{array}]$

Langkah 4.3

Find the null space when $λ = 15.81232455$ .

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Langkah 4.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} - 14.81232455 & 2 & 6 & 8 & 0 \\ 1 & 40.18767544 & 2 & 85 & 0 \\ 16 & 7 & - 6.81232455 & 3 & 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 4.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{- 14.81232455}$ to make the entry at $1, 1$ a $1$ .

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Langkah 4.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{- 14.81232455}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 14.81232455}{- 14.81232455} & \frac{2}{- 14.81232455} & \frac{6}{- 14.81232455} & \frac{8}{- 14.81232455} & \frac{0}{- 14.81232455} \\ 1 & 40.18767544 & 2 & 85 & 0 \\ 16 & 7 & - 6.81232455 & 3 & 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 1 & 40.18767544 & 2 & 85 & 0 \\ 16 & 7 & - 6.81232455 & 3 & 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 4.3.2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 1 - 1 & 40.18767544 + 0.13502269 & 2 + 0.40506808 & 85 + 0.54009078 & 0 - 0 \\ 16 & 7 & - 6.81232455 & 3 & 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 40.32269813 & 2.40506808 & 85.54009078 & 0 \\ 16 & 7 & - 6.81232455 & 3 & 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2.3

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 4.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 40.32269813 & 2.40506808 & 85.54009078 & 0 \\ 16 - 16 \cdot 1 & 7 - 16 \cdot - 0.13502269 & - 6.81232455 - 16 \cdot - 0.40506808 & 3 - 16 \cdot - 0.54009078 & 0 - 16 \cdot 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 40.32269813 & 2.40506808 & 85.54009078 & 0 \\ 0 & 9.16036314 & - 0.33123513 & 11.64145256 & 0 \\ 4 & 5 & 7 & 83.18767544 & 0 \end{array}]$

Langkah 4.3.2.4

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Langkah 4.3.2.4.1

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 40.32269813 & 2.40506808 & 85.54009078 & 0 \\ 0 & 9.16036314 & - 0.33123513 & 11.64145256 & 0 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot - 0.13502269 & 7 - 4 \cdot - 0.40506808 & 83.18767544 - 4 \cdot - 0.54009078 & 0 - 4 \cdot 0 \end{array}]$

Langkah 4.3.2.4.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 40.32269813 & 2.40506808 & 85.54009078 & 0 \\ 0 & 9.16036314 & - 0.33123513 & 11.64145256 & 0 \\ 0 & 5.54009078 & 8.62027235 & 85.34803858 & 0 \end{array}]$

Langkah 4.3.2.5

Multiply each element of $R_{2}$ by $\frac{1}{40.32269813}$ to make the entry at $2, 2$ a $1$ .

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Langkah 4.3.2.5.1

Multiply each element of $R_{2}$ by $\frac{1}{40.32269813}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ \frac{0}{40.32269813} & \frac{40.32269813}{40.32269813} & \frac{2.40506808}{40.32269813} & \frac{85.54009078}{40.32269813} & \frac{0}{40.32269813} \\ 0 & 9.16036314 & - 0.33123513 & 11.64145256 & 0 \\ 0 & 5.54009078 & 8.62027235 & 85.34803858 & 0 \end{array}]$

Langkah 4.3.2.5.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 9.16036314 & - 0.33123513 & 11.64145256 & 0 \\ 0 & 5.54009078 & 8.62027235 & 85.34803858 & 0 \end{array}]$

Langkah 4.3.2.6

Perform the row operation $R_{3} = R_{3} - 9.16036314 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 4.3.2.6.1

Perform the row operation $R_{3} = R_{3} - 9.16036314 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 - 9.16036314 \cdot 0 & 9.16036314 - 9.16036314 \cdot 1 & - 0.33123513 - 9.16036314 \cdot 0.05964551 & 11.64145256 - 9.16036314 \cdot 2.12138807 & 0 - 9.16036314 \cdot 0 \\ 0 & 5.54009078 & 8.62027235 & 85.34803858 & 0 \end{array}]$

Langkah 4.3.2.6.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & - 0.87760971 & - 7.79123252 & 0 \\ 0 & 5.54009078 & 8.62027235 & 85.34803858 & 0 \end{array}]$

Langkah 4.3.2.7

Perform the row operation $R_{4} = R_{4} - 5.54009078 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Langkah 4.3.2.7.1

Perform the row operation $R_{4} = R_{4} - 5.54009078 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & - 0.87760971 & - 7.79123252 & 0 \\ 0 - 5.54009078 \cdot 0 & 5.54009078 - 5.54009078 \cdot 1 & 8.62027235 - 5.54009078 \cdot 0.05964551 & 85.34803858 - 5.54009078 \cdot 2.12138807 & 0 - 5.54009078 \cdot 0 \end{array}]$

Langkah 4.3.2.7.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & - 0.87760971 & - 7.79123252 & 0 \\ 0 & 0 & 8.28983078 & 73.59535608 & 0 \end{array}]$

Langkah 4.3.2.8

Multiply each element of $R_{3}$ by $\frac{1}{- 0.87760971}$ to make the entry at $3, 3$ a $1$ .

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Langkah 4.3.2.8.1

Multiply each element of $R_{3}$ by $\frac{1}{- 0.87760971}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ \frac{0}{- 0.87760971} & \frac{0}{- 0.87760971} & \frac{- 0.87760971}{- 0.87760971} & \frac{- 7.79123252}{- 0.87760971} & \frac{0}{- 0.87760971} \\ 0 & 0 & 8.28983078 & 73.59535608 & 0 \end{array}]$

Langkah 4.3.2.8.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & 1 & 8.87778749 & 0 \\ 0 & 0 & 8.28983078 & 73.59535608 & 0 \end{array}]$

Langkah 4.3.2.9

Perform the row operation $R_{4} = R_{4} - 8.28983078 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Langkah 4.3.2.9.1

Perform the row operation $R_{4} = R_{4} - 8.28983078 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & 1 & 8.87778749 & 0 \\ 0 - 8.28983078 \cdot 0 & 0 - 8.28983078 \cdot 0 & 8.28983078 - 8.28983078 \cdot 1 & 73.59535608 - 8.28983078 \cdot 8.87778749 & 0 - 8.28983078 \cdot 0 \end{array}]$

Langkah 4.3.2.9.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & 1 & 8.87778749 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Langkah 4.3.2.10

Multiply each element of $R_{4}$ by $\frac{1}{- 1.36594866 \cdot 10^{- 12}}$ to make the entry at $4, 4$ a $1$ .

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Langkah 4.3.2.10.1

Multiply each element of $R_{4}$ by $\frac{1}{- 1.36594866 \cdot 10^{- 12}}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & 1 & 8.87778749 & 0 \\ \frac{0}{- 1.36594866 \cdot 10^{- 12}} & \frac{0}{- 1.36594866 \cdot 10^{- 12}} & \frac{0}{- 1.36594866 \cdot 10^{- 12}} & \frac{0}{- 1.36594866 \cdot 10^{- 12}} & \frac{0}{- 1.36594866 \cdot 10^{- 12}} \end{array}]$

Langkah 4.3.2.10.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & 1 & 8.87778749 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.11

Perform the row operation $R_{3} = R_{3} - 8.87778749 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Langkah 4.3.2.11.1

Perform the row operation $R_{3} = R_{3} - 8.87778749 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 - 8.87778749 \cdot 0 & 0 - 8.87778749 \cdot 0 & 1 - 8.87778749 \cdot 0 & 8.87778749 - 8.87778749 \cdot 1 & 0 - 8.87778749 \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.11.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 2.12138807 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.12

Perform the row operation $R_{2} = R_{2} - 2.12138807 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Langkah 4.3.2.12.1

Perform the row operation $R_{2} = R_{2} - 2.12138807 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 - 2.12138807 \cdot 0 & 1 - 2.12138807 \cdot 0 & 0.05964551 - 2.12138807 \cdot 0 & 2.12138807 - 2.12138807 \cdot 1 & 0 - 2.12138807 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.12.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & - 0.54009078 & 0 \\ 0 & 1 & 0.05964551 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.13

Perform the row operation $R_{1} = R_{1} + 0.54009078 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Langkah 4.3.2.13.1

Perform the row operation $R_{1} = R_{1} + 0.54009078 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 0.54009078 \cdot 0 & - 0.13502269 + 0.54009078 \cdot 0 & - 0.40506808 + 0.54009078 \cdot 0 & - 0.54009078 + 0.54009078 \cdot 1 & 0 + 0.54009078 \cdot 0 \\ 0 & 1 & 0.05964551 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.13.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & 0 & 0 \\ 0 & 1 & 0.05964551 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.14

Perform the row operation $R_{2} = R_{2} - 0.05964551 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Langkah 4.3.2.14.1

Perform the row operation $R_{2} = R_{2} - 0.05964551 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & 0 & 0 \\ 0 - 0.05964551 \cdot 0 & 1 - 0.05964551 \cdot 0 & 0.05964551 - 0.05964551 \cdot 1 & 0 - 0.05964551 \cdot 0 & 0 - 0.05964551 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.14.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.13502269 & - 0.40506808 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.15

Perform the row operation $R_{1} = R_{1} + 0.40506808 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Langkah 4.3.2.15.1

Perform the row operation $R_{1} = R_{1} + 0.40506808 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 0.40506808 \cdot 0 & - 0.13502269 + 0.40506808 \cdot 0 & - 0.40506808 + 0.40506808 \cdot 1 & 0 + 0.40506808 \cdot 0 & 0 + 0.40506808 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.15.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 0.13502269 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.16

Perform the row operation $R_{1} = R_{1} + 0.13502269 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 4.3.2.16.1

Perform the row operation $R_{1} = R_{1} + 0.13502269 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 0.13502269 \cdot 0 & - 0.13502269 + 0.13502269 \cdot 1 & 0 + 0.13502269 \cdot 0 & 0 + 0.13502269 \cdot 0 & 0 + 0.13502269 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.2.16.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 4.3.3

Use the result matrix to declare the final solution to the system of equations.

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

Langkah 4.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]$

Langkah 4.3.5

Write as a solution set.

${[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]}$

Langkah 5

Find the eigenvector using the eigenvalue $λ = 45.88268919$ .

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Langkah 5.1

Substitusikan nilai-nilai yang diketahui ke dalam rumusnya.

$N ([\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] - 45.88268919 [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Langkah 5.2

Sederhanakan.

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Langkah 5.2.1

Sederhanakan setiap suku.

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Langkah 5.2.1.1

Kalikan $- 45.88268919$ dengan setiap elemen di dalam matriks tersebut.

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2

Sederhanakan setiap elemen dalam matriks.

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Langkah 5.2.1.2.1

Kalikan $- 45.88268919$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.2

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.3

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.4

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.5

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.6

Kalikan $- 45.88268919$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.7

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.8

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.9

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.10

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ 0 & 0 & - 45.88268919 \cdot 1 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.11

Kalikan $- 45.88268919$ dengan $1$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ 0 & 0 & - 45.88268919 & - 45.88268919 \cdot 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.12

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ 0 & 0 & - 45.88268919 & 0 \\ - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.13

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ 0 & 0 & - 45.88268919 & 0 \\ 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.14

Kalikan $- 45.88268919$ dengan $0$ .

$[\begin{array}{c} 1 & 2 & 6 & 8 \\ 1 & 56 & 2 & 85 \\ 16 & 7 & 9 & 3 \\ 4 & 5 & 7 & 99 \end{array}] + [\begin{array}{c} - 45.88268919 & 0 & 0 & 0 \\ 0 & - 45.88268919 & 0 & 0 \\ 0 & 0 & - 45.88268919 & 0 \\ 0 & 0 & - 45.88268919 \cdot 0 & - 45.88268919 \cdot 1 \end{array}]$

Langkah 5.2.1.2.15

Kalikan $- 45.88268919$ dengan $0$ .

Langkah 5.2.1.2.16

Kalikan $- 45.88268919$ dengan $1$ .

Langkah 5.2.2

Tambahkan elemen yang seletak.

$[\begin{array}{c} 1 - 45.88268919 & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - 45.88268919 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3

Simplify each element.

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Langkah 5.2.3.1

Kurangi $45.88268919$ dengan $1$ .

$[\begin{array}{c} - 44.88268919 & 2 + 0 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - 45.88268919 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.2

Tambahkan $2$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 + 0 & 8 + 0 \\ 1 + 0 & 56 - 45.88268919 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.3

Tambahkan $6$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 + 0 \\ 1 + 0 & 56 - 45.88268919 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.4

Tambahkan $8$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 + 0 & 56 - 45.88268919 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.5

Tambahkan $1$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 56 - 45.88268919 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.6

Kurangi $45.88268919$ dengan $56$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 + 0 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.7

Tambahkan $2$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 + 0 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.8

Tambahkan $85$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 + 0 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.9

Tambahkan $16$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 + 0 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.10

Tambahkan $7$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & 9 - 45.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.11

Kurangi $45.88268919$ dengan $9$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & - 36.88268919 & 3 + 0 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.12

Tambahkan $3$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & - 36.88268919 & 3 \\ 4 + 0 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.13

Tambahkan $4$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & - 36.88268919 & 3 \\ 4 & 5 + 0 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.14

Tambahkan $5$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & - 36.88268919 & 3 \\ 4 & 5 & 7 + 0 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.15

Tambahkan $7$ dan $0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & - 36.88268919 & 3 \\ 4 & 5 & 7 & 99 - 45.88268919 \end{array}]$

Langkah 5.2.3.16

Kurangi $45.88268919$ dengan $99$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 \\ 1 & 10.1173108 & 2 & 85 \\ 16 & 7 & - 36.88268919 & 3 \\ 4 & 5 & 7 & 53.1173108 \end{array}]$

Langkah 5.3

Find the null space when $λ = 45.88268919$ .

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Langkah 5.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} - 44.88268919 & 2 & 6 & 8 & 0 \\ 1 & 10.1173108 & 2 & 85 & 0 \\ 16 & 7 & - 36.88268919 & 3 & 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 5.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{- 44.88268919}$ to make the entry at $1, 1$ a $1$ .

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Langkah 5.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{- 44.88268919}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 44.88268919}{- 44.88268919} & \frac{2}{- 44.88268919} & \frac{6}{- 44.88268919} & \frac{8}{- 44.88268919} & \frac{0}{- 44.88268919} \\ 1 & 10.1173108 & 2 & 85 & 0 \\ 16 & 7 & - 36.88268919 & 3 & 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 1 & 10.1173108 & 2 & 85 & 0 \\ 16 & 7 & - 36.88268919 & 3 & 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 5.3.2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 1 - 1 & 10.1173108 + 0.0445606 & 2 + 0.13368182 & 85 + 0.17824243 & 0 - 0 \\ 16 & 7 & - 36.88268919 & 3 & 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 10.16187141 & 2.13368182 & 85.17824243 & 0 \\ 16 & 7 & - 36.88268919 & 3 & 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2.3

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 5.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 10.16187141 & 2.13368182 & 85.17824243 & 0 \\ 16 - 16 \cdot 1 & 7 - 16 \cdot - 0.0445606 & - 36.88268919 - 16 \cdot - 0.13368182 & 3 - 16 \cdot - 0.17824243 & 0 - 16 \cdot 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 10.16187141 & 2.13368182 & 85.17824243 & 0 \\ 0 & 7.71296975 & - 34.74377992 & 5.85187902 & 0 \\ 4 & 5 & 7 & 53.1173108 & 0 \end{array}]$

Langkah 5.3.2.4

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Langkah 5.3.2.4.1

Perform the row operation $R_{4} = R_{4} - 4 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 10.16187141 & 2.13368182 & 85.17824243 & 0 \\ 0 & 7.71296975 & - 34.74377992 & 5.85187902 & 0 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot - 0.0445606 & 7 - 4 \cdot - 0.13368182 & 53.1173108 - 4 \cdot - 0.17824243 & 0 - 4 \cdot 0 \end{array}]$

Langkah 5.3.2.4.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 10.16187141 & 2.13368182 & 85.17824243 & 0 \\ 0 & 7.71296975 & - 34.74377992 & 5.85187902 & 0 \\ 0 & 5.17824243 & 7.53472731 & 53.83028055 & 0 \end{array}]$

Langkah 5.3.2.5

Multiply each element of $R_{2}$ by $\frac{1}{10.16187141}$ to make the entry at $2, 2$ a $1$ .

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Langkah 5.3.2.5.1

Multiply each element of $R_{2}$ by $\frac{1}{10.16187141}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ \frac{0}{10.16187141} & \frac{10.16187141}{10.16187141} & \frac{2.13368182}{10.16187141} & \frac{85.17824243}{10.16187141} & \frac{0}{10.16187141} \\ 0 & 7.71296975 & - 34.74377992 & 5.85187902 & 0 \\ 0 & 5.17824243 & 7.53472731 & 53.83028055 & 0 \end{array}]$

Langkah 5.3.2.5.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 7.71296975 & - 34.74377992 & 5.85187902 & 0 \\ 0 & 5.17824243 & 7.53472731 & 53.83028055 & 0 \end{array}]$

Langkah 5.3.2.6

Perform the row operation $R_{3} = R_{3} - 7.71296975 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Langkah 5.3.2.6.1

Perform the row operation $R_{3} = R_{3} - 7.71296975 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 - 7.71296975 \cdot 0 & 7.71296975 - 7.71296975 \cdot 1 & - 34.74377992 - 7.71296975 \cdot 0.20996937 & 5.85187902 - 7.71296975 \cdot 8.38214133 & 0 - 7.71296975 \cdot 0 \\ 0 & 5.17824243 & 7.53472731 & 53.83028055 & 0 \end{array}]$

Langkah 5.3.2.6.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & - 36.36326739 & - 58.79932362 & 0 \\ 0 & 5.17824243 & 7.53472731 & 53.83028055 & 0 \end{array}]$

Langkah 5.3.2.7

Perform the row operation $R_{4} = R_{4} - 5.17824243 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Langkah 5.3.2.7.1

Perform the row operation $R_{4} = R_{4} - 5.17824243 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & - 36.36326739 & - 58.79932362 & 0 \\ 0 - 5.17824243 \cdot 0 & 5.17824243 - 5.17824243 \cdot 1 & 7.53472731 - 5.17824243 \cdot 0.20996937 & 53.83028055 - 5.17824243 \cdot 8.38214133 & 0 - 5.17824243 \cdot 0 \end{array}]$

Langkah 5.3.2.7.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & - 36.36326739 & - 58.79932362 & 0 \\ 0 & 0 & 6.44745496 & 10.42552054 & 0 \end{array}]$

Langkah 5.3.2.8

Multiply each element of $R_{3}$ by $\frac{1}{- 36.36326739}$ to make the entry at $3, 3$ a $1$ .

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Langkah 5.3.2.8.1

Multiply each element of $R_{3}$ by $\frac{1}{- 36.36326739}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ \frac{0}{- 36.36326739} & \frac{0}{- 36.36326739} & \frac{- 36.36326739}{- 36.36326739} & \frac{- 58.79932362}{- 36.36326739} & \frac{0}{- 36.36326739} \\ 0 & 0 & 6.44745496 & 10.42552054 & 0 \end{array}]$

Langkah 5.3.2.8.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & 1 & 1.6169978 & 0 \\ 0 & 0 & 6.44745496 & 10.42552054 & 0 \end{array}]$

Langkah 5.3.2.9

Perform the row operation $R_{4} = R_{4} - 6.44745496 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Langkah 5.3.2.9.1

Perform the row operation $R_{4} = R_{4} - 6.44745496 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & 1 & 1.6169978 & 0 \\ 0 - 6.44745496 \cdot 0 & 0 - 6.44745496 \cdot 0 & 6.44745496 - 6.44745496 \cdot 1 & 10.42552054 - 6.44745496 \cdot 1.6169978 & 0 - 6.44745496 \cdot 0 \end{array}]$

Langkah 5.3.2.9.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & 1 & 1.6169978 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Langkah 5.3.2.10

Multiply each element of $R_{4}$ by $\frac{1}{- 2.57631995 \cdot 10^{- 13}}$ to make the entry at $4, 4$ a $1$ .

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Langkah 5.3.2.10.1

Multiply each element of $R_{4}$ by $\frac{1}{- 2.57631995 \cdot 10^{- 13}}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & 1 & 1.6169978 & 0 \\ \frac{0}{- 2.57631995 \cdot 10^{- 13}} & \frac{0}{- 2.57631995 \cdot 10^{- 13}} & \frac{0}{- 2.57631995 \cdot 10^{- 13}} & \frac{0}{- 2.57631995 \cdot 10^{- 13}} & \frac{0}{- 2.57631995 \cdot 10^{- 13}} \end{array}]$

Langkah 5.3.2.10.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & 1 & 1.6169978 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.11

Perform the row operation $R_{3} = R_{3} - 1.6169978 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Langkah 5.3.2.11.1

Perform the row operation $R_{3} = R_{3} - 1.6169978 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 - 1.6169978 \cdot 0 & 0 - 1.6169978 \cdot 0 & 1 - 1.6169978 \cdot 0 & 1.6169978 - 1.6169978 \cdot 1 & 0 - 1.6169978 \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.11.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 8.38214133 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.12

Perform the row operation $R_{2} = R_{2} - 8.38214133 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Langkah 5.3.2.12.1

Perform the row operation $R_{2} = R_{2} - 8.38214133 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 - 8.38214133 \cdot 0 & 1 - 8.38214133 \cdot 0 & 0.20996937 - 8.38214133 \cdot 0 & 8.38214133 - 8.38214133 \cdot 1 & 0 - 8.38214133 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.12.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & - 0.17824243 & 0 \\ 0 & 1 & 0.20996937 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.13

Perform the row operation $R_{1} = R_{1} + 0.17824243 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Langkah 5.3.2.13.1

Perform the row operation $R_{1} = R_{1} + 0.17824243 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 0.17824243 \cdot 0 & - 0.0445606 + 0.17824243 \cdot 0 & - 0.13368182 + 0.17824243 \cdot 0 & - 0.17824243 + 0.17824243 \cdot 1 & 0 + 0.17824243 \cdot 0 \\ 0 & 1 & 0.20996937 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.13.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & 0 & 0 \\ 0 & 1 & 0.20996937 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.14

Perform the row operation $R_{2} = R_{2} - 0.20996937 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Langkah 5.3.2.14.1

Perform the row operation $R_{2} = R_{2} - 0.20996937 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & 0 & 0 \\ 0 - 0.20996937 \cdot 0 & 1 - 0.20996937 \cdot 0 & 0.20996937 - 0.20996937 \cdot 1 & 0 - 0.20996937 \cdot 0 & 0 - 0.20996937 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.14.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 0.0445606 & - 0.13368182 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.15

Perform the row operation $R_{1} = R_{1} + 0.13368182 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Langkah 5.3.2.15.1

Perform the row operation $R_{1} = R_{1} + 0.13368182 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 0.13368182 \cdot 0 & - 0.0445606 + 0.13368182 \cdot 0 & - 0.13368182 + 0.13368182 \cdot 1 & 0 + 0.13368182 \cdot 0 & 0 + 0.13368182 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.15.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 0.0445606 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.16

Perform the row operation $R_{1} = R_{1} + 0.0445606 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 5.3.2.16.1

Perform the row operation $R_{1} = R_{1} + 0.0445606 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 0.0445606 \cdot 0 & - 0.0445606 + 0.0445606 \cdot 1 & 0 + 0.0445606 \cdot 0 & 0 + 0.0445606 \cdot 0 & 0 + 0.0445606 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.2.16.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Langkah 5.3.3

Use the result matrix to declare the final solution to the system of equations.

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

Langkah 5.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]$

Langkah 5.3.5

Write as a solution set.

${[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]}$

Langkah 6

The eigenspace of $A$ is the list of the vector space for each eigenvalue.

${[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}]}$