Matematika Berhingga Contoh

$[\begin{array}{c} 0 & 0 & 0 & - 1 \\ 0 & 0 & 2 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \end{array}]$

Langkah 1

Find the determinant.

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Langkah 1.1

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 1.1.1

Consider the corresponding sign chart.

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

Langkah 1.1.2

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

Langkah 1.1.3

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

$| \begin{array}{c} 0 & 2 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 0 \end{array} |$

Langkah 1.1.4

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

$0 | \begin{array}{c} 0 & 2 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 0 \end{array} |$

Langkah 1.1.5

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

$| \begin{array}{c} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} |$

Langkah 1.1.6

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

$0 | \begin{array}{c} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} |$

Langkah 1.1.7

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

$| \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.1.8

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

$0 | \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.1.9

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

$| \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.1.10

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

$1 | \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.1.11

Add the terms together.

$0 | \begin{array}{c} 0 & 2 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 0 \end{array} | + 0 | \begin{array}{c} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} | + 1 | \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.2

Kalikan $0$ dengan $| \begin{array}{c} 0 & 2 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 0 \end{array} |$ .

$0 + 0 | \begin{array}{c} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} | + 0 | \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} | + 1 | \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.3

Kalikan $0$ dengan $| \begin{array}{c} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} |$ .

$0 + 0 + 0 | \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} | + 1 | \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.4

Kalikan $0$ dengan $| \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$ .

$0 + 0 + 0 + 1 | \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$

Langkah 1.5

Evaluasi $| \begin{array}{c} 0 & 0 & 2 \\ 1 & 0 & 0 \\ 0 & 3 & 0 \end{array} |$ .

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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.

$| \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Langkah 1.5.1.4

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

$0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Langkah 1.5.1.5

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

$| \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array} |$

Langkah 1.5.1.6

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

$0 | \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array} |$

Langkah 1.5.1.7

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

$| \begin{array}{c} 1 & 0 \\ 0 & 3 \end{array} |$

Langkah 1.5.1.8

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

$2 | \begin{array}{c} 1 & 0 \\ 0 & 3 \end{array} |$

Langkah 1.5.1.9

Add the terms together.

$0 + 0 + 0 + 1 (0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} | + 0 | \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array} | + 2 | \begin{array}{c} 1 & 0 \\ 0 & 3 \end{array} |)$

Langkah 1.5.2

Kalikan $0$ dengan $| \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$ .

$0 + 0 + 0 + 1 (0 + 0 | \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array} | + 2 | \begin{array}{c} 1 & 0 \\ 0 & 3 \end{array} |)$

Langkah 1.5.3

Kalikan $0$ dengan $| \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array} |$ .

$0 + 0 + 0 + 1 (0 + 0 + 2 | \begin{array}{c} 1 & 0 \\ 0 & 3 \end{array} |)$

Langkah 1.5.4

Evaluasi $| \begin{array}{c} 1 & 0 \\ 0 & 3 \end{array} |$ .

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Langkah 1.5.4.1

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

$0 + 0 + 0 + 1 (0 + 0 + 2 (1 \cdot 3 + 0 \cdot 0))$

Langkah 1.5.4.2

Sederhanakan determinannya.

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Langkah 1.5.4.2.1

Sederhanakan setiap suku.

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Langkah 1.5.4.2.1.1

Kalikan $3$ dengan $1$ .

$0 + 0 + 0 + 1 (0 + 0 + 2 (3 + 0 \cdot 0))$

Langkah 1.5.4.2.1.2

Kalikan $0$ dengan $0$ .

$0 + 0 + 0 + 1 (0 + 0 + 2 (3 + 0))$

Langkah 1.5.4.2.2

Tambahkan $3$ dan $0$ .

$0 + 0 + 0 + 1 (0 + 0 + 2 \cdot 3)$

Langkah 1.5.5

Sederhanakan determinannya.

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Langkah 1.5.5.1

Kalikan $2$ dengan $3$ .

$0 + 0 + 0 + 1 (0 + 0 + 6)$

Langkah 1.5.5.2

Tambahkan $0$ dan $0$ .

$0 + 0 + 0 + 1 (0 + 6)$

Langkah 1.5.5.3

Tambahkan $0$ dan $6$ .

$0 + 0 + 0 + 1 \cdot 6$

Langkah 1.6

Sederhanakan determinannya.

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Langkah 1.6.1

Kalikan $6$ dengan $1$ .

$0 + 0 + 0 + 6$

Langkah 1.6.2

Tambahkan $0$ dan $0$ .

$0 + 0 + 6$

Langkah 1.6.3

Tambahkan $0$ dan $0$ .

$0 + 6$

Langkah 1.6.4

Tambahkan $0$ dan $6$ .

$6$

Langkah 2

Since the determinant is non-zero, the inverse exists.

Langkah 3

Set up a $4 \times 8$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Langkah 4

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 4.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Langkah 4.2

Swap $R_{4}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \end{array}]$

Langkah 4.3

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

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Langkah 4.3.1

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \frac{0}{3} & \frac{3}{3} & \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{1}{3} \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \end{array}]$

Langkah 4.3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \end{array}]$

Langkah 4.4

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \end{array}]$

Langkah 4.5

Multiply each element of $R_{3}$ by $\frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

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Langkah 4.5.1

Multiply each element of $R_{3}$ by $\frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{0}{2} & \frac{0}{2} & \frac{1}{2} & \frac{0}{2} & \frac{0}{2} \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \end{array}]$

Langkah 4.5.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & - 1 & 1 & 0 & 0 & 0 \end{array}]$

Langkah 4.6

Multiply each element of $R_{4}$ by $- 1$ to make the entry at $4, 4$ a $1$ .

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Langkah 4.6.1

Multiply each element of $R_{4}$ by $- 1$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ - 0 & - 0 & - 0 & - - 1 & - 1 \cdot 1 & - 0 & - 0 & - 0 \end{array}]$

Langkah 4.6.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & 0 & 0 & 0 \end{array}]$

Langkah 5

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{1}{3} \\ 0 & \frac{1}{2} & 0 & 0 \\ - 1 & 0 & 0 & 0 \end{array}]$