Linear Algebra Examples

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}]$

Step 1

Find the eigenvalues.

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Step 1.1

Set up the formula to find the characteristic equation $p (λ)$ .

$p (λ) = determinant (A - λ I_{4})$

Step 1.2

The identity matrix or unit matrix of size $4$ is the $4 \times 4$ square matrix with ones on the main diagonal and zeros elsewhere.

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

Step 1.3

Substitute the known values into $p (λ) = determinant (A - λ I_{4})$ .

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Step 1.3.1

Substitute $[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] - λ I_{4})$

Step 1.3.2

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

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $- λ$ by each element of the matrix.

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.5

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.5.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.5.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.9

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.9.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.9.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.10

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.10.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.10.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.11

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.12

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.12.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.12.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.13

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.13.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.13.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.14

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.14.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.14.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.15

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.15.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.15.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.16

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}])$

Step 1.4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 + 0 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 - λ & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Add $5$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 - λ & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.2

Add $5$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 + 0 \\ 6 + 0 & 5 - λ & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.3

Add $5$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 + 0 & 5 - λ & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.4

Add $6$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.5

Add $5$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.6

Add $6$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 + 0 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.7

Add $7$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 & 8 + 0 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.8

Add $8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 & 8 & 9 - λ & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.9

Add $8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 & 8 & 9 - λ & 8 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.10

Add $6$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 & 8 & 9 - λ & 8 \\ 6 & 7 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.11

Add $7$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 & 8 & 9 - λ & 8 \\ 6 & 7 & 8 + 0 & 9 - λ \end{array}]$

Step 1.4.3.12

Add $8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 5 - λ & 5 & 5 & 5 \\ 6 & 5 - λ & 5 & 6 \\ 7 & 8 & 9 - λ & 8 \\ 6 & 7 & 8 & 9 - λ \end{array}]$

Step 1.5

Find the determinant.

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Step 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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Step 1.5.1.1

Consider the corresponding sign chart.

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

Step 1.5.1.2

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

Step 1.5.1.3

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

$| \begin{array}{c} 5 - λ & 5 & 6 \\ 8 & 9 - λ & 8 \\ 7 & 8 & 9 - λ \end{array} |$

Step 1.5.1.4

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

$(5 - λ) | \begin{array}{c} 5 - λ & 5 & 6 \\ 8 & 9 - λ & 8 \\ 7 & 8 & 9 - λ \end{array} |$

Step 1.5.1.5

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

$| \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} |$

Step 1.5.1.6

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

$- 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} |$

Step 1.5.1.7

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

$| \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} |$

Step 1.5.1.8

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

$5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} |$

Step 1.5.1.9

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

$| \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.1.10

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

$- 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.1.11

Add the terms together.

$p (λ) = (5 - λ) | \begin{array}{c} 5 - λ & 5 & 6 \\ 8 & 9 - λ & 8 \\ 7 & 8 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2

Evaluate $| \begin{array}{c} 5 - λ & 5 & 6 \\ 8 & 9 - λ & 8 \\ 7 & 8 & 9 - λ \end{array} |$ .

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Step 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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Step 1.5.2.1.1

Consider the corresponding sign chart.

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

Step 1.5.2.1.2

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

Step 1.5.2.1.3

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

$| \begin{array}{c} 9 - λ & 8 \\ 8 & 9 - λ \end{array} |$

Step 1.5.2.1.4

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

$(5 - λ) | \begin{array}{c} 9 - λ & 8 \\ 8 & 9 - λ \end{array} |$

Step 1.5.2.1.5

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

$| \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} |$

Step 1.5.2.1.6

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

$- 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} |$

Step 1.5.2.1.7

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

$| \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |$

Step 1.5.2.1.8

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

$6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |$

Step 1.5.2.1.9

Add the terms together.

$p (λ) = (5 - λ) ((5 - λ) | \begin{array}{c} 9 - λ & 8 \\ 8 & 9 - λ \end{array} | - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2

Evaluate $| \begin{array}{c} 9 - λ & 8 \\ 8 & 9 - λ \end{array} |$ .

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Step 1.5.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) ((5 - λ) ((9 - λ) (9 - λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2

Simplify the determinant.

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Step 1.5.2.2.2.1

Simplify each term.

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Step 1.5.2.2.2.1.1

Expand $(9 - λ) (9 - λ)$ using the FOIL Method.

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Step 1.5.2.2.2.1.1.1

Apply the distributive property.

$p (λ) = (5 - λ) ((5 - λ) (9 (9 - λ) - λ (9 - λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.1.2

Apply the distributive property.

$p (λ) = (5 - λ) ((5 - λ) (9 \cdot 9 + 9 (- λ) - λ (9 - λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.1.3

Apply the distributive property.

$p (λ) = (5 - λ) ((5 - λ) (9 \cdot 9 + 9 (- λ) - λ \cdot 9 - λ (- λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2

Simplify and combine like terms.

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Step 1.5.2.2.2.1.2.1

Simplify each term.

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Step 1.5.2.2.2.1.2.1.1

Multiply $9$ by $9$ .

$p (λ) = (5 - λ) ((5 - λ) (81 + 9 (- λ) - λ \cdot 9 - λ (- λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.2

Multiply $- 1$ by $9$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - λ \cdot 9 - λ (- λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.3

Multiply $9$ by $- 1$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - 9 λ - λ (- λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - 9 λ - 1 \cdot - 1 λ \cdot λ - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.5

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.2.2.2.1.2.1.5.1

Move $λ$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - 9 λ - 1 \cdot - 1 (λ \cdot λ) - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - 9 λ - 1 \cdot - 1 λ^{2} - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - 9 λ + 1 λ^{2} - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 9 λ - 9 λ + λ^{2} - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.2.2

Subtract $9 λ$ from $- 9 λ$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 18 λ + λ^{2} - 8 \cdot 8) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.1.3

Multiply $- 8$ by $8$ .

$p (λ) = (5 - λ) ((5 - λ) (81 - 18 λ + λ^{2} - 64) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.2

Subtract $64$ from $81$ .

$p (λ) = (5 - λ) ((5 - λ) (- 18 λ + λ^{2} + 17) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.2.2.3

Reorder $- 18 λ$ and $λ^{2}$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.3

Evaluate $| \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} |$ .

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Step 1.5.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (8 (9 - λ) - 7 \cdot 8) + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.3.2

Simplify the determinant.

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Step 1.5.2.3.2.1

Simplify each term.

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Step 1.5.2.3.2.1.1

Apply the distributive property.

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (8 \cdot 9 + 8 (- λ) - 7 \cdot 8) + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.3.2.1.2

Multiply $8$ by $9$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (72 + 8 (- λ) - 7 \cdot 8) + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.3.2.1.3

Multiply $- 1$ by $8$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (72 - 8 λ - 7 \cdot 8) + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.3.2.1.4

Multiply $- 7$ by $8$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (72 - 8 λ - 56) + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.3.2.2

Subtract $56$ from $72$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4

Evaluate $| \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |$ .

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Step 1.5.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (8 \cdot 8 - 7 (9 - λ))) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4.2

Simplify the determinant.

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Step 1.5.2.4.2.1

Simplify each term.

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Step 1.5.2.4.2.1.1

Multiply $8$ by $8$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (64 - 7 (9 - λ))) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4.2.1.2

Apply the distributive property.

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (64 - 7 \cdot 9 - 7 (- λ))) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4.2.1.3

Multiply $- 7$ by $9$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (64 - 63 - 7 (- λ))) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4.2.1.4

Multiply $- 1$ by $- 7$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (64 - 63 + 7 λ)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4.2.2

Subtract $63$ from $64$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (1 + 7 λ)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.4.2.3

Reorder $1$ and $7 λ$ .

$p (λ) = (5 - λ) ((5 - λ) (λ^{2} - 18 λ + 17) - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5

Simplify the determinant.

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Step 1.5.2.5.1

Simplify each term.

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Step 1.5.2.5.1.1

Expand $(5 - λ) (λ^{2} - 18 λ + 17)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = (5 - λ) (5 λ^{2} + 5 (- 18 λ) + 5 \cdot 17 - λ \cdot λ^{2} - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2

Simplify each term.

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Step 1.5.2.5.1.2.1

Multiply $- 18$ by $5$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 5 \cdot 17 - λ \cdot λ^{2} - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.2

Multiply $5$ by $17$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ \cdot λ^{2} - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.3

Multiply $λ$ by $λ^{2}$ by adding the exponents.

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Step 1.5.2.5.1.2.3.1

Move $λ^{2}$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - (λ^{2} λ) - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.3.2

Multiply $λ^{2}$ by $λ$ .

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Step 1.5.2.5.1.2.3.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - (λ^{2} λ^{1}) - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{2 + 1} - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.3.3

Add $2$ and $1$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{3} - λ (- 18 λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.4

Rewrite using the commutative property of multiplication.

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{3} - 1 \cdot - 18 λ \cdot λ - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.5

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.2.5.1.2.5.1

Move $λ$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{3} - 1 \cdot - 18 (λ \cdot λ) - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.5.2

Multiply $λ$ by $λ$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{3} - 1 \cdot - 18 λ^{2} - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.6

Multiply $- 1$ by $- 18$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{3} + 18 λ^{2} - λ \cdot 17 - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.2.7

Multiply $17$ by $- 1$ .

$p (λ) = (5 - λ) (5 λ^{2} - 90 λ + 85 - λ^{3} + 18 λ^{2} - 17 λ - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.3

Add $5 λ^{2}$ and $18 λ^{2}$ .

$p (λ) = (5 - λ) (23 λ^{2} - 90 λ + 85 - λ^{3} - 17 λ - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.4

Subtract $17 λ$ from $- 90 λ$ .

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} - 5 (- 8 λ + 16) + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.5

Apply the distributive property.

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} - 5 (- 8 λ) - 5 \cdot 16 + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.6

Multiply $- 8$ by $- 5$ .

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} + 40 λ - 5 \cdot 16 + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.7

Multiply $- 5$ by $16$ .

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} + 40 λ - 80 + 6 (7 λ + 1)) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.8

Apply the distributive property.

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} + 40 λ - 80 + 6 (7 λ) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.9

Multiply $7$ by $6$ .

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} + 40 λ - 80 + 42 λ + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.1.10

Multiply $6$ by $1$ .

$p (λ) = (5 - λ) (23 λ^{2} - 107 λ + 85 - λ^{3} + 40 λ - 80 + 42 λ + 6) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.2

Add $- 107 λ$ and $40 λ$ .

$p (λ) = (5 - λ) (23 λ^{2} - 67 λ + 85 - λ^{3} - 80 + 42 λ + 6) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.3

Add $- 67 λ$ and $42 λ$ .

$p (λ) = (5 - λ) (23 λ^{2} - 25 λ + 85 - λ^{3} - 80 + 6) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.4

Subtract $80$ from $85$ .

$p (λ) = (5 - λ) (23 λ^{2} - 25 λ - λ^{3} + 5 + 6) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.5

Add $5$ and $6$ .

$p (λ) = (5 - λ) (23 λ^{2} - 25 λ - λ^{3} + 11) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.6

Move $- 25 λ$ .

$p (λ) = (5 - λ) (23 λ^{2} - λ^{3} - 25 λ + 11) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.2.5.7

Reorder $23 λ^{2}$ and $- λ^{3}$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 | \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} | + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3

Evaluate $| \begin{array}{c} 6 & 5 & 6 \\ 7 & 9 - λ & 8 \\ 6 & 8 & 9 - λ \end{array} |$ .

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Step 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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Step 1.5.3.1.1

Consider the corresponding sign chart.

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

Step 1.5.3.1.2

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

Step 1.5.3.1.3

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

$| \begin{array}{c} 5 & 6 \\ 8 & 9 - λ \end{array} |$

Step 1.5.3.1.4

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

$- 7 | \begin{array}{c} 5 & 6 \\ 8 & 9 - λ \end{array} |$

Step 1.5.3.1.5

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

$| \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} |$

Step 1.5.3.1.6

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

$(9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} |$

Step 1.5.3.1.7

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

$| \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |$

Step 1.5.3.1.8

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

$- 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |$

Step 1.5.3.1.9

Add the terms together.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 | \begin{array}{c} 5 & 6 \\ 8 & 9 - λ \end{array} | + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.2

Evaluate $| \begin{array}{c} 5 & 6 \\ 8 & 9 - λ \end{array} |$ .

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Step 1.5.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (5 (9 - λ) - 8 \cdot 6) + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.2.2

Simplify the determinant.

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Step 1.5.3.2.2.1

Simplify each term.

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Step 1.5.3.2.2.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (5 \cdot 9 + 5 (- λ) - 8 \cdot 6) + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.2.2.1.2

Multiply $5$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (45 + 5 (- λ) - 8 \cdot 6) + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.2.2.1.3

Multiply $- 1$ by $5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (45 - 5 λ - 8 \cdot 6) + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.2.2.1.4

Multiply $- 8$ by $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (45 - 5 λ - 48) + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.2.2.2

Subtract $48$ from $45$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) | \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} | - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.3

Evaluate $| \begin{array}{c} 6 & 6 \\ 6 & 9 - λ \end{array} |$ .

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Step 1.5.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (6 (9 - λ) - 6 \cdot 6) - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.3.2

Simplify the determinant.

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Step 1.5.3.3.2.1

Simplify each term.

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Step 1.5.3.3.2.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (6 \cdot 9 + 6 (- λ) - 6 \cdot 6) - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.3.2.1.2

Multiply $6$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (54 + 6 (- λ) - 6 \cdot 6) - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.3.2.1.3

Multiply $- 1$ by $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (54 - 6 λ - 6 \cdot 6) - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.3.2.1.4

Multiply $- 6$ by $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (54 - 6 λ - 36) - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.3.2.2

Subtract $36$ from $54$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (- 6 λ + 18) - 8 | \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.4

Evaluate $| \begin{array}{c} 6 & 5 \\ 6 & 8 \end{array} |$ .

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Step 1.5.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (- 6 λ + 18) - 8 (6 \cdot 8 - 6 \cdot 5)) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.4.2

Simplify the determinant.

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Step 1.5.3.4.2.1

Simplify each term.

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Step 1.5.3.4.2.1.1

Multiply $6$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (- 6 λ + 18) - 8 (48 - 6 \cdot 5)) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.4.2.1.2

Multiply $- 6$ by $5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (- 6 λ + 18) - 8 (48 - 30)) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.4.2.2

Subtract $30$ from $48$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ - 3) + (9 - λ) (- 6 λ + 18) - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5

Simplify the determinant.

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Step 1.5.3.5.1

Simplify each term.

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Step 1.5.3.5.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 7 (- 5 λ) - 7 \cdot - 3 + (9 - λ) (- 6 λ + 18) - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.2

Multiply $- 5$ by $- 7$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ - 7 \cdot - 3 + (9 - λ) (- 6 λ + 18) - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.3

Multiply $- 7$ by $- 3$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 + (9 - λ) (- 6 λ + 18) - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.4

Expand $(9 - λ) (- 6 λ + 18)$ using the FOIL Method.

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Step 1.5.3.5.1.4.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 + 9 (- 6 λ + 18) - λ (- 6 λ + 18) - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.4.2

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 + 9 (- 6 λ) + 9 \cdot 18 - λ (- 6 λ + 18) - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.4.3

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 + 9 (- 6 λ) + 9 \cdot 18 - λ (- 6 λ) - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5

Simplify and combine like terms.

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Step 1.5.3.5.1.5.1

Simplify each term.

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Step 1.5.3.5.1.5.1.1

Multiply $- 6$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 9 \cdot 18 - λ (- 6 λ) - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.1.2

Multiply $9$ by $18$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 162 - λ (- 6 λ) - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.1.3

Rewrite using the commutative property of multiplication.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 162 - 1 \cdot - 6 λ \cdot λ - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.1.4

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.3.5.1.5.1.4.1

Move $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 162 - 1 \cdot - 6 (λ \cdot λ) - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.1.4.2

Multiply $λ$ by $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 162 - 1 \cdot - 6 λ^{2} - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.1.5

Multiply $- 1$ by $- 6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 162 + 6 λ^{2} - λ \cdot 18 - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.1.6

Multiply $18$ by $- 1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 54 λ + 162 + 6 λ^{2} - 18 λ - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.5.2

Subtract $18 λ$ from $- 54 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 72 λ + 162 + 6 λ^{2} - 8 \cdot 18) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.1.6

Multiply $- 8$ by $18$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (35 λ + 21 - 72 λ + 162 + 6 λ^{2} - 144) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.2

Subtract $72 λ$ from $35 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 37 λ + 21 + 162 + 6 λ^{2} - 144) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.3

Add $21$ and $162$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 37 λ + 183 + 6 λ^{2} - 144) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.4

Subtract $144$ from $183$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (- 37 λ + 6 λ^{2} + 39) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.3.5.5

Reorder $- 37 λ$ and $6 λ^{2}$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 | \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} | - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4

Evaluate $| \begin{array}{c} 6 & 5 - λ & 6 \\ 7 & 8 & 8 \\ 6 & 7 & 9 - λ \end{array} |$ .

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Step 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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Step 1.5.4.1.1

Consider the corresponding sign chart.

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

Step 1.5.4.1.2

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

Step 1.5.4.1.3

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

$| \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} |$

Step 1.5.4.1.4

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

$6 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} |$

Step 1.5.4.1.5

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

$| \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} |$

Step 1.5.4.1.6

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

$- (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} |$

Step 1.5.4.1.7

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

$| \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |$

Step 1.5.4.1.8

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

$6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |$

Step 1.5.4.1.9

Add the terms together.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 | \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} | - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.2

Evaluate $| \begin{array}{c} 8 & 8 \\ 7 & 9 - λ \end{array} |$ .

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Step 1.5.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (8 (9 - λ) - 7 \cdot 8) - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.2.2

Simplify the determinant.

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Step 1.5.4.2.2.1

Simplify each term.

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Step 1.5.4.2.2.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (8 \cdot 9 + 8 (- λ) - 7 \cdot 8) - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.2.2.1.2

Multiply $8$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (72 + 8 (- λ) - 7 \cdot 8) - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.2.2.1.3

Multiply $- 1$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (72 - 8 λ - 7 \cdot 8) - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.2.2.1.4

Multiply $- 7$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (72 - 8 λ - 56) - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.2.2.2

Subtract $56$ from $72$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) | \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} | + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.3

Evaluate $| \begin{array}{c} 7 & 8 \\ 6 & 9 - λ \end{array} |$ .

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Step 1.5.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (7 (9 - λ) - 6 \cdot 8) + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.3.2

Simplify the determinant.

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Step 1.5.4.3.2.1

Simplify each term.

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Step 1.5.4.3.2.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (7 \cdot 9 + 7 (- λ) - 6 \cdot 8) + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.3.2.1.2

Multiply $7$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (63 + 7 (- λ) - 6 \cdot 8) + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.3.2.1.3

Multiply $- 1$ by $7$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (63 - 7 λ - 6 \cdot 8) + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.3.2.1.4

Multiply $- 6$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (63 - 7 λ - 48) + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.3.2.2

Subtract $48$ from $63$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (- 7 λ + 15) + 6 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.4

Evaluate $| \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |$ .

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Step 1.5.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (- 7 λ + 15) + 6 (7 \cdot 7 - 6 \cdot 8)) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.4.2

Simplify the determinant.

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Step 1.5.4.4.2.1

Simplify each term.

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Step 1.5.4.4.2.1.1

Multiply $7$ by $7$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (- 7 λ + 15) + 6 (49 - 6 \cdot 8)) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.4.2.1.2

Multiply $- 6$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (- 7 λ + 15) + 6 (49 - 48)) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.4.2.2

Subtract $48$ from $49$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ + 16) - (5 - λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5

Simplify the determinant.

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Step 1.5.4.5.1

Simplify each term.

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Step 1.5.4.5.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (6 (- 8 λ) + 6 \cdot 16 - (5 - λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.2

Multiply $- 8$ by $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 6 \cdot 16 - (5 - λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.3

Multiply $6$ by $16$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 - (5 - λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.4

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + (- 1 \cdot 5 - - λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.5

Multiply $- 1$ by $5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + (- 5 - - λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.6

Multiply $- - λ$ .

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Step 1.5.4.5.1.6.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + (- 5 + 1 λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.6.2

Multiply $λ$ by $1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + (- 5 + λ) (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.7

Expand $(- 5 + λ) (- 7 λ + 15)$ using the FOIL Method.

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Step 1.5.4.5.1.7.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 - 5 (- 7 λ + 15) + λ (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.7.2

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 - 5 (- 7 λ) - 5 \cdot 15 + λ (- 7 λ + 15) + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.7.3

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 - 5 (- 7 λ) - 5 \cdot 15 + λ (- 7 λ) + λ \cdot 15 + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8

Simplify and combine like terms.

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Step 1.5.4.5.1.8.1

Simplify each term.

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Step 1.5.4.5.1.8.1.1

Multiply $- 7$ by $- 5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 35 λ - 5 \cdot 15 + λ (- 7 λ) + λ \cdot 15 + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8.1.2

Multiply $- 5$ by $15$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 35 λ - 75 + λ (- 7 λ) + λ \cdot 15 + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8.1.3

Rewrite using the commutative property of multiplication.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 35 λ - 75 - 7 λ \cdot λ + λ \cdot 15 + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8.1.4

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.4.5.1.8.1.4.1

Move $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 35 λ - 75 - 7 (λ \cdot λ) + λ \cdot 15 + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8.1.4.2

Multiply $λ$ by $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 35 λ - 75 - 7 λ^{2} + λ \cdot 15 + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8.1.5

Move $15$ to the left of $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 35 λ - 75 - 7 λ^{2} + 15 λ + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.8.2

Add $35 λ$ and $15 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 50 λ - 75 - 7 λ^{2} + 6 \cdot 1) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.1.9

Multiply $6$ by $1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 48 λ + 96 + 50 λ - 75 - 7 λ^{2} + 6) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.2

Add $- 48 λ$ and $50 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (2 λ + 96 - 75 - 7 λ^{2} + 6) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.3

Subtract $75$ from $96$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (2 λ + 21 - 7 λ^{2} + 6) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.4

Add $21$ and $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (2 λ - 7 λ^{2} + 27) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.4.5.5

Reorder $2 λ$ and $- 7 λ^{2}$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 | \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$

Step 1.5.5

Evaluate $| \begin{array}{c} 6 & 5 - λ & 5 \\ 7 & 8 & 9 - λ \\ 6 & 7 & 8 \end{array} |$ .

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Step 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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Step 1.5.5.1.1

Consider the corresponding sign chart.

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

Step 1.5.5.1.2

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

Step 1.5.5.1.3

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

$| \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |$

Step 1.5.5.1.4

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

$6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |$

Step 1.5.5.1.5

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

$| \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} |$

Step 1.5.5.1.6

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

$- (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} |$

Step 1.5.5.1.7

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

$| \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |$

Step 1.5.5.1.8

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

$5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |$

Step 1.5.5.1.9

Add the terms together.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 | \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} | - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2

Evaluate $| \begin{array}{c} 8 & 9 - λ \\ 7 & 8 \end{array} |$ .

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Step 1.5.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (8 \cdot 8 - 7 (9 - λ)) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2.2

Simplify the determinant.

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Step 1.5.5.2.2.1

Simplify each term.

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Step 1.5.5.2.2.1.1

Multiply $8$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (64 - 7 (9 - λ)) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2.2.1.2

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (64 - 7 \cdot 9 - 7 (- λ)) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2.2.1.3

Multiply $- 7$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (64 - 63 - 7 (- λ)) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2.2.1.4

Multiply $- 1$ by $- 7$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (64 - 63 + 7 λ) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2.2.2

Subtract $63$ from $64$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (1 + 7 λ) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.2.2.3

Reorder $1$ and $7 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) | \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} | + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3

Evaluate $| \begin{array}{c} 7 & 9 - λ \\ 6 & 8 \end{array} |$ .

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Step 1.5.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (7 \cdot 8 - 6 (9 - λ)) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3.2

Simplify the determinant.

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Step 1.5.5.3.2.1

Simplify each term.

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Step 1.5.5.3.2.1.1

Multiply $7$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (56 - 6 (9 - λ)) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3.2.1.2

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (56 - 6 \cdot 9 - 6 (- λ)) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3.2.1.3

Multiply $- 6$ by $9$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (56 - 54 - 6 (- λ)) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3.2.1.4

Multiply $- 1$ by $- 6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (56 - 54 + 6 λ) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3.2.2

Subtract $54$ from $56$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (2 + 6 λ) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.3.2.3

Reorder $2$ and $6 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (6 λ + 2) + 5 | \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |)$

Step 1.5.5.4

Evaluate $| \begin{array}{c} 7 & 8 \\ 6 & 7 \end{array} |$ .

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Step 1.5.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (6 λ + 2) + 5 (7 \cdot 7 - 6 \cdot 8))$

Step 1.5.5.4.2

Simplify the determinant.

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Step 1.5.5.4.2.1

Simplify each term.

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Step 1.5.5.4.2.1.1

Multiply $7$ by $7$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (6 λ + 2) + 5 (49 - 6 \cdot 8))$

Step 1.5.5.4.2.1.2

Multiply $- 6$ by $8$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (6 λ + 2) + 5 (49 - 48))$

Step 1.5.5.4.2.2

Subtract $48$ from $49$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ + 1) - (5 - λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5

Simplify the determinant.

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Step 1.5.5.5.1

Simplify each term.

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Step 1.5.5.5.1.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 (7 λ) + 6 \cdot 1 - (5 - λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.2

Multiply $7$ by $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 \cdot 1 - (5 - λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.3

Multiply $6$ by $1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - (5 - λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.4

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 + (- 1 \cdot 5 - - λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.5

Multiply $- 1$ by $5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 + (- 5 - - λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.6

Multiply $- - λ$ .

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Step 1.5.5.5.1.6.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 + (- 5 + 1 λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.6.2

Multiply $λ$ by $1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 + (- 5 + λ) (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.7

Expand $(- 5 + λ) (6 λ + 2)$ using the FOIL Method.

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Step 1.5.5.5.1.7.1

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 5 (6 λ + 2) + λ (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.7.2

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 5 (6 λ) - 5 \cdot 2 + λ (6 λ + 2) + 5 \cdot 1)$

Step 1.5.5.5.1.7.3

Apply the distributive property.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 5 (6 λ) - 5 \cdot 2 + λ (6 λ) + λ \cdot 2 + 5 \cdot 1)$

Step 1.5.5.5.1.8

Simplify and combine like terms.

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Step 1.5.5.5.1.8.1

Simplify each term.

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Step 1.5.5.5.1.8.1.1

Multiply $6$ by $- 5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 30 λ - 5 \cdot 2 + λ (6 λ) + λ \cdot 2 + 5 \cdot 1)$

Step 1.5.5.5.1.8.1.2

Multiply $- 5$ by $2$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 30 λ - 10 + λ (6 λ) + λ \cdot 2 + 5 \cdot 1)$

Step 1.5.5.5.1.8.1.3

Rewrite using the commutative property of multiplication.

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 30 λ - 10 + 6 λ \cdot λ + λ \cdot 2 + 5 \cdot 1)$

Step 1.5.5.5.1.8.1.4

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.5.5.1.8.1.4.1

Move $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 30 λ - 10 + 6 (λ \cdot λ) + λ \cdot 2 + 5 \cdot 1)$

Step 1.5.5.5.1.8.1.4.2

Multiply $λ$ by $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 30 λ - 10 + 6 λ^{2} + λ \cdot 2 + 5 \cdot 1)$

Step 1.5.5.5.1.8.1.5

Move $2$ to the left of $λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 30 λ - 10 + 6 λ^{2} + 2 λ + 5 \cdot 1)$

Step 1.5.5.5.1.8.2

Add $- 30 λ$ and $2 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 28 λ - 10 + 6 λ^{2} + 5 \cdot 1)$

Step 1.5.5.5.1.9

Multiply $5$ by $1$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (42 λ + 6 - 28 λ - 10 + 6 λ^{2} + 5)$

Step 1.5.5.5.2

Subtract $28 λ$ from $42 λ$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (14 λ + 6 - 10 + 6 λ^{2} + 5)$

Step 1.5.5.5.3

Subtract $10$ from $6$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (14 λ - 4 + 6 λ^{2} + 5)$

Step 1.5.5.5.4

Add $- 4$ and $5$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (14 λ + 6 λ^{2} + 1)$

Step 1.5.5.5.5

Reorder $14 λ$ and $6 λ^{2}$ .

$p (λ) = (5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11) - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6

Simplify the determinant.

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Step 1.5.6.1

Simplify each term.

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Step 1.5.6.1.1

Expand $(5 - λ) (- λ^{3} + 23 λ^{2} - 25 λ + 11)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = 5 (- λ^{3}) + 5 (23 λ^{2}) + 5 (- 25 λ) + 5 \cdot 11 - λ (- λ^{3}) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2

Simplify each term.

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Step 1.5.6.1.2.1

Multiply $- 1$ by $5$ .

$p (λ) = - 5 λ^{3} + 5 (23 λ^{2}) + 5 (- 25 λ) + 5 \cdot 11 - λ (- λ^{3}) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.2

Multiply $23$ by $5$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} + 5 (- 25 λ) + 5 \cdot 11 - λ (- λ^{3}) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.3

Multiply $- 25$ by $5$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 5 \cdot 11 - λ (- λ^{3}) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.4

Multiply $5$ by $11$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 - λ (- λ^{3}) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.5

Rewrite using the commutative property of multiplication.

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 - 1 \cdot - 1 λ \cdot λ^{3} - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.6

Multiply $λ$ by $λ^{3}$ by adding the exponents.

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Step 1.5.6.1.2.6.1

Move $λ^{3}$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 - 1 \cdot - 1 (λ^{3} λ) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.6.2

Multiply $λ^{3}$ by $λ$ .

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Step 1.5.6.1.2.6.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 - 1 \cdot - 1 (λ^{3} λ^{1}) - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 - 1 \cdot - 1 λ^{3 + 1} - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.6.3

Add $3$ and $1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 - 1 \cdot - 1 λ^{4} - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.7

Multiply $- 1$ by $- 1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + 1 λ^{4} - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.8

Multiply $λ^{4}$ by $1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - λ (23 λ^{2}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.9

Rewrite using the commutative property of multiplication.

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 1 \cdot 23 λ \cdot λ^{2} - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.10

Multiply $λ$ by $λ^{2}$ by adding the exponents.

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Step 1.5.6.1.2.10.1

Move $λ^{2}$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 1 \cdot 23 (λ^{2} λ) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.10.2

Multiply $λ^{2}$ by $λ$ .

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Step 1.5.6.1.2.10.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 1 \cdot 23 (λ^{2} λ^{1}) - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 1 \cdot 23 λ^{2 + 1} - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.10.3

Add $2$ and $1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 1 \cdot 23 λ^{3} - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.11

Multiply $- 1$ by $23$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 23 λ^{3} - λ (- 25 λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.12

Rewrite using the commutative property of multiplication.

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 23 λ^{3} - 1 \cdot - 25 λ \cdot λ - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.13

Multiply $λ$ by $λ$ by adding the exponents.

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Step 1.5.6.1.2.13.1

Move $λ$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 23 λ^{3} - 1 \cdot - 25 (λ \cdot λ) - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.13.2

Multiply $λ$ by $λ$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 23 λ^{3} - 1 \cdot - 25 λ^{2} - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.14

Multiply $- 1$ by $- 25$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 23 λ^{3} + 25 λ^{2} - λ \cdot 11 - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.2.15

Multiply $11$ by $- 1$ .

$p (λ) = - 5 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} - 23 λ^{3} + 25 λ^{2} - 11 λ - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.3

Subtract $23 λ^{3}$ from $- 5 λ^{3}$ .

$p (λ) = - 28 λ^{3} + 115 λ^{2} - 125 λ + 55 + λ^{4} + 25 λ^{2} - 11 λ - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.4

Add $115 λ^{2}$ and $25 λ^{2}$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 125 λ + 55 + λ^{4} - 11 λ - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.5

Subtract $11 λ$ from $- 125 λ$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 5 (6 λ^{2} - 37 λ + 39) + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.6

Apply the distributive property.

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 5 (6 λ^{2}) - 5 (- 37 λ) - 5 \cdot 39 + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.7

Simplify.

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Step 1.5.6.1.7.1

Multiply $6$ by $- 5$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} - 5 (- 37 λ) - 5 \cdot 39 + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.7.2

Multiply $- 37$ by $- 5$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 5 \cdot 39 + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.7.3

Multiply $- 5$ by $39$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 + 5 (- 7 λ^{2} + 2 λ + 27) - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.8

Apply the distributive property.

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 + 5 (- 7 λ^{2}) + 5 (2 λ) + 5 \cdot 27 - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.9

Simplify.

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Step 1.5.6.1.9.1

Multiply $- 7$ by $5$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 5 (2 λ) + 5 \cdot 27 - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.9.2

Multiply $2$ by $5$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 10 λ + 5 \cdot 27 - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.9.3

Multiply $5$ by $27$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 10 λ + 135 - 5 (6 λ^{2} + 14 λ + 1)$

Step 1.5.6.1.10

Apply the distributive property.

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 10 λ + 135 - 5 (6 λ^{2}) - 5 (14 λ) - 5 \cdot 1$

Step 1.5.6.1.11

Simplify.

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Step 1.5.6.1.11.1

Multiply $6$ by $- 5$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 10 λ + 135 - 30 λ^{2} - 5 (14 λ) - 5 \cdot 1$

Step 1.5.6.1.11.2

Multiply $14$ by $- 5$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 10 λ + 135 - 30 λ^{2} - 70 λ - 5 \cdot 1$

Step 1.5.6.1.11.3

Multiply $- 5$ by $1$ .

$p (λ) = - 28 λ^{3} + 140 λ^{2} - 136 λ + 55 + λ^{4} - 30 λ^{2} + 185 λ - 195 - 35 λ^{2} + 10 λ + 135 - 30 λ^{2} - 70 λ - 5$

Step 1.5.6.2

Subtract $30 λ^{2}$ from $140 λ^{2}$ .

$p (λ) = - 28 λ^{3} + 110 λ^{2} - 136 λ + 55 + λ^{4} + 185 λ - 195 - 35 λ^{2} + 10 λ + 135 - 30 λ^{2} - 70 λ - 5$

Step 1.5.6.3

Subtract $35 λ^{2}$ from $110 λ^{2}$ .

$p (λ) = - 28 λ^{3} + 75 λ^{2} - 136 λ + 55 + λ^{4} + 185 λ - 195 + 10 λ + 135 - 30 λ^{2} - 70 λ - 5$

Step 1.5.6.4

Subtract $30 λ^{2}$ from $75 λ^{2}$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} - 136 λ + 55 + λ^{4} + 185 λ - 195 + 10 λ + 135 - 70 λ - 5$

Step 1.5.6.5

Add $- 136 λ$ and $185 λ$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} + 49 λ + 55 + λ^{4} - 195 + 10 λ + 135 - 70 λ - 5$

Step 1.5.6.6

Add $49 λ$ and $10 λ$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} + 59 λ + 55 + λ^{4} - 195 + 135 - 70 λ - 5$

Step 1.5.6.7

Subtract $70 λ$ from $59 λ$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} - 11 λ + 55 + λ^{4} - 195 + 135 - 5$

Step 1.5.6.8

Subtract $195$ from $55$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} - 11 λ + λ^{4} - 140 + 135 - 5$

Step 1.5.6.9

Add $- 140$ and $135$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} - 11 λ + λ^{4} - 5 - 5$

Step 1.5.6.10

Subtract $5$ from $- 5$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} - 11 λ + λ^{4} - 10$

Step 1.5.6.11

Move $- 11 λ$ .

$p (λ) = - 28 λ^{3} + 45 λ^{2} + λ^{4} - 11 λ - 10$

Step 1.5.6.12

Move $45 λ^{2}$ .

$p (λ) = - 28 λ^{3} + λ^{4} + 45 λ^{2} - 11 λ - 10$

Step 1.5.6.13

Reorder $- 28 λ^{3}$ and $λ^{4}$ .

$p (λ) = λ^{4} - 28 λ^{3} + 45 λ^{2} - 11 λ - 10$

Step 1.6

Set the characteristic polynomial equal to $0$ to find the eigenvalues $λ$ .

$λ^{4} - 28 λ^{3} + 45 λ^{2} - 11 λ - 10 = 0$

Step 1.7

Solve for $λ$ .

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Step 1.7.1

Graph each side of the equation. The solution is the x-value of the point of intersection.

$λ \approx - 0.33891265, 26.30579577$

Step 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.

$ε_{A} = N (A - λ I_{4})$

Step 3

Find the eigenvector using the eigenvalue $λ = - 0.33891265$ .

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Step 3.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + 0.33891265 [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $0.33891265$ by each element of the matrix.

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply $0.33891265$ by $1$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.5

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.6

Multiply $0.33891265$ by $1$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.7

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.8

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.9

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.10

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0 & 0.33891265 \cdot 1 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.11

Multiply $0.33891265$ by $1$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0 & 0.33891265 & 0.33891265 \cdot 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.12

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0 & 0.33891265 & 0 \\ 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.13

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0 & 0.33891265 & 0 \\ 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.14

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0 & 0.33891265 & 0 \\ 0 & 0 & 0.33891265 \cdot 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.15

Multiply $0.33891265$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} 0.33891265 & 0 & 0 & 0 \\ 0 & 0.33891265 & 0 & 0 \\ 0 & 0 & 0.33891265 & 0 \\ 0 & 0 & 0 & 0.33891265 \cdot 1 \end{array}]$

Step 3.2.1.2.16

Multiply $0.33891265$ by $1$ .

Step 3.2.2

Add the corresponding elements.

$[\begin{array}{c} 5 + 0.33891265 & 5 + 0 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 + 0.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3

Simplify each element.

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Step 3.2.3.1

Add $5$ and $0.33891265$ .

$[\begin{array}{c} 5.33891265 & 5 + 0 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 + 0.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.2

Add $5$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 + 0.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.3

Add $5$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 + 0 \\ 6 + 0 & 5 + 0.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.4

Add $5$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 + 0 & 5 + 0.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.5

Add $6$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5 + 0.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.6

Add $5$ and $0.33891265$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.7

Add $5$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.8

Add $6$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 + 0 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.9

Add $7$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 + 0 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.10

Add $8$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9 + 0.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.11

Add $9$ and $0.33891265$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9.33891265 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.12

Add $8$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9.33891265 & 8 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.13

Add $6$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9.33891265 & 8 \\ 6 & 7 + 0 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.14

Add $7$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9.33891265 & 8 \\ 6 & 7 & 8 + 0 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.15

Add $8$ and $0$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9.33891265 & 8 \\ 6 & 7 & 8 & 9 + 0.33891265 \end{array}]$

Step 3.2.3.16

Add $9$ and $0.33891265$ .

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 \\ 6 & 5.33891265 & 5 & 6 \\ 7 & 8 & 9.33891265 & 8 \\ 6 & 7 & 8 & 9.33891265 \end{array}]$

Step 3.3

Find the null space when $λ = - 0.33891265$ .

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Step 3.3.1

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

$[\begin{array}{c} 5.33891265 & 5 & 5 & 5 & 0 \\ 6 & 5.33891265 & 5 & 6 & 0 \\ 7 & 8 & 9.33891265 & 8 & 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2

Find the reduced row echelon form.

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Step 3.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{5.33891265}$ to make the entry at $1, 1$ a $1$ .

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Step 3.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{5.33891265}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{5.33891265}{5.33891265} & \frac{5}{5.33891265} & \frac{5}{5.33891265} & \frac{5}{5.33891265} & \frac{0}{5.33891265} \\ 6 & 5.33891265 & 5 & 6 & 0 \\ 7 & 8 & 9.33891265 & 8 & 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 6 & 5.33891265 & 5 & 6 & 0 \\ 7 & 8 & 9.33891265 & 8 & 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.3.2.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 6 - 6 \cdot 1 & 5.33891265 - 6 \cdot 0.93652028 & 5 - 6 \cdot 0.93652028 & 6 - 6 \cdot 0.93652028 & 0 - 6 \cdot 0 \\ 7 & 8 & 9.33891265 & 8 & 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & - 0.28020905 & - 0.6191217 & 0.38087829 & 0 \\ 7 & 8 & 9.33891265 & 8 & 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2.3

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & - 0.28020905 & - 0.6191217 & 0.38087829 & 0 \\ 7 - 7 \cdot 1 & 8 - 7 \cdot 0.93652028 & 9.33891265 - 7 \cdot 0.93652028 & 8 - 7 \cdot 0.93652028 & 0 - 7 \cdot 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & - 0.28020905 & - 0.6191217 & 0.38087829 & 0 \\ 0 & 1.444358 & 2.78327065 & 1.444358 & 0 \\ 6 & 7 & 8 & 9.33891265 & 0 \end{array}]$

Step 3.3.2.4

Perform the row operation $R_{4} = R_{4} - 6 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 3.3.2.4.1

Perform the row operation $R_{4} = R_{4} - 6 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & - 0.28020905 & - 0.6191217 & 0.38087829 & 0 \\ 0 & 1.444358 & 2.78327065 & 1.444358 & 0 \\ 6 - 6 \cdot 1 & 7 - 6 \cdot 0.93652028 & 8 - 6 \cdot 0.93652028 & 9.33891265 - 6 \cdot 0.93652028 & 0 - 6 \cdot 0 \end{array}]$

Step 3.3.2.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & - 0.28020905 & - 0.6191217 & 0.38087829 & 0 \\ 0 & 1.444358 & 2.78327065 & 1.444358 & 0 \\ 0 & 1.38087829 & 2.38087829 & 3.71979094 & 0 \end{array}]$

Step 3.3.2.5

Multiply each element of $R_{2}$ by $\frac{1}{- 0.28020905}$ to make the entry at $2, 2$ a $1$ .

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Step 3.3.2.5.1

Multiply each element of $R_{2}$ by $\frac{1}{- 0.28020905}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ \frac{0}{- 0.28020905} & \frac{- 0.28020905}{- 0.28020905} & \frac{- 0.6191217}{- 0.28020905} & \frac{0.38087829}{- 0.28020905} & \frac{0}{- 0.28020905} \\ 0 & 1.444358 & 2.78327065 & 1.444358 & 0 \\ 0 & 1.38087829 & 2.38087829 & 3.71979094 & 0 \end{array}]$

Step 3.3.2.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 1.444358 & 2.78327065 & 1.444358 & 0 \\ 0 & 1.38087829 & 2.38087829 & 3.71979094 & 0 \end{array}]$

Step 3.3.2.6

Perform the row operation $R_{3} = R_{3} - 1.444358 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 3.3.2.6.1

Perform the row operation $R_{3} = R_{3} - 1.444358 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 - 1.444358 \cdot 0 & 1.444358 - 1.444358 \cdot 1 & 2.78327065 - 1.444358 \cdot 2.20949928 & 1.444358 - 1.444358 \cdot - 1.35926474 & 0 - 1.444358 \cdot 0 \\ 0 & 1.38087829 & 2.38087829 & 3.71979094 & 0 \end{array}]$

Step 3.3.2.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & - 0.40803732 & 3.40762291 & 0 \\ 0 & 1.38087829 & 2.38087829 & 3.71979094 & 0 \end{array}]$

Step 3.3.2.7

Perform the row operation $R_{4} = R_{4} - 1.38087829 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 3.3.2.7.1

Perform the row operation $R_{4} = R_{4} - 1.38087829 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & - 0.40803732 & 3.40762291 & 0 \\ 0 - 1.38087829 \cdot 0 & 1.38087829 - 1.38087829 \cdot 1 & 2.38087829 - 1.38087829 \cdot 2.20949928 & 3.71979094 - 1.38087829 \cdot - 1.35926474 & 0 - 1.38087829 \cdot 0 \end{array}]$

Step 3.3.2.7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & - 0.40803732 & 3.40762291 & 0 \\ 0 & 0 & - 0.6701713 & 5.59677011 & 0 \end{array}]$

Step 3.3.2.8

Multiply each element of $R_{3}$ by $\frac{1}{- 0.40803732}$ to make the entry at $3, 3$ a $1$ .

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Step 3.3.2.8.1

Multiply each element of $R_{3}$ by $\frac{1}{- 0.40803732}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ \frac{0}{- 0.40803732} & \frac{0}{- 0.40803732} & \frac{- 0.40803732}{- 0.40803732} & \frac{3.40762291}{- 0.40803732} & \frac{0}{- 0.40803732} \\ 0 & 0 & - 0.6701713 & 5.59677011 & 0 \end{array}]$

Step 3.3.2.8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & 1 & - 8.35125293 & 0 \\ 0 & 0 & - 0.6701713 & 5.59677011 & 0 \end{array}]$

Step 3.3.2.9

Perform the row operation $R_{4} = R_{4} + 0.6701713 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 3.3.2.9.1

Perform the row operation $R_{4} = R_{4} + 0.6701713 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & 1 & - 8.35125293 & 0 \\ 0 + 0.6701713 \cdot 0 & 0 + 0.6701713 \cdot 0 & - 0.6701713 + 0.6701713 \cdot 1 & 5.59677011 + 0.6701713 \cdot - 8.35125293 & 0 + 0.6701713 \cdot 0 \end{array}]$

Step 3.3.2.9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & 1 & - 8.35125293 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.10

Multiply each element of $R_{4}$ by $\frac{1}{- 1.16703367 \cdot 10^{- 10}}$ to make the entry at $4, 4$ a $1$ .

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Step 3.3.2.10.1

Multiply each element of $R_{4}$ by $\frac{1}{- 1.16703367 \cdot 10^{- 10}}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & 1 & - 8.35125293 & 0 \\ \frac{0}{- 1.16703367 \cdot 10^{- 10}} & \frac{0}{- 1.16703367 \cdot 10^{- 10}} & \frac{0}{- 1.16703367 \cdot 10^{- 10}} & \frac{0}{- 1.16703367 \cdot 10^{- 10}} & \frac{0}{- 1.16703367 \cdot 10^{- 10}} \end{array}]$

Step 3.3.2.10.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & 1 & - 8.35125293 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.11

Perform the row operation $R_{3} = R_{3} + 8.35125293 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 3.3.2.11.1

Perform the row operation $R_{3} = R_{3} + 8.35125293 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 + 8.35125293 \cdot 0 & 0 + 8.35125293 \cdot 0 & 1 + 8.35125293 \cdot 0 & - 8.35125293 + 8.35125293 \cdot 1 & 0 + 8.35125293 \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.11.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & - 1.35926474 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.12

Perform the row operation $R_{2} = R_{2} + 1.35926474 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 3.3.2.12.1

Perform the row operation $R_{2} = R_{2} + 1.35926474 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 + 1.35926474 \cdot 0 & 1 + 1.35926474 \cdot 0 & 2.20949928 + 1.35926474 \cdot 0 & - 1.35926474 + 1.35926474 \cdot 1 & 0 + 1.35926474 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.12.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0.93652028 & 0 \\ 0 & 1 & 2.20949928 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.13

Perform the row operation $R_{1} = R_{1} - 0.93652028 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Step 3.3.2.13.1

Perform the row operation $R_{1} = R_{1} - 0.93652028 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 0.93652028 \cdot 0 & 0.93652028 - 0.93652028 \cdot 0 & 0.93652028 - 0.93652028 \cdot 0 & 0.93652028 - 0.93652028 \cdot 1 & 0 - 0.93652028 \cdot 0 \\ 0 & 1 & 2.20949928 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.13.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0 & 0 \\ 0 & 1 & 2.20949928 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.14

Perform the row operation $R_{2} = R_{2} - 2.20949928 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 3.3.2.14.1

Perform the row operation $R_{2} = R_{2} - 2.20949928 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0 & 0 \\ 0 - 2.20949928 \cdot 0 & 1 - 2.20949928 \cdot 0 & 2.20949928 - 2.20949928 \cdot 1 & 0 - 2.20949928 \cdot 0 & 0 - 2.20949928 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.14.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0.93652028 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.15

Perform the row operation $R_{1} = R_{1} - 0.93652028 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 3.3.2.15.1

Perform the row operation $R_{1} = R_{1} - 0.93652028 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.93652028 \cdot 0 & 0.93652028 - 0.93652028 \cdot 0 & 0.93652028 - 0.93652028 \cdot 1 & 0 - 0.93652028 \cdot 0 & 0 - 0.93652028 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0.93652028 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.16

Perform the row operation $R_{1} = R_{1} - 0.93652028 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.2.16.1

Perform the row operation $R_{1} = R_{1} - 0.93652028 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.93652028 \cdot 0 & 0.93652028 - 0.93652028 \cdot 1 & 0 - 0.93652028 \cdot 0 & 0 - 0.93652028 \cdot 0 & 0 - 0.93652028 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.16.2

Simplify $R_{1}$ .

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

Step 3.3.3

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

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

Step 3.3.4

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

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

Step 3.3.5

Write as a solution set.

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

Step 4

Find the eigenvector using the eigenvalue $λ = 26.30579577$ .

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Step 4.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] - 26.30579577 [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 4.2

Simplify.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $- 26.30579577$ by each element of the matrix.

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2

Simplify each element in the matrix.

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Step 4.2.1.2.1

Multiply $- 26.30579577$ by $1$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.2

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.3

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.4

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.5

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.6

Multiply $- 26.30579577$ by $1$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.7

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.8

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.9

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.10

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & 0 & - 26.30579577 \cdot 1 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.11

Multiply $- 26.30579577$ by $1$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & 0 & - 26.30579577 & - 26.30579577 \cdot 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.12

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & 0 & - 26.30579577 & 0 \\ - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.13

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & 0 & - 26.30579577 & 0 \\ 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.14

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & 0 & - 26.30579577 & 0 \\ 0 & 0 & - 26.30579577 \cdot 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.15

Multiply $- 26.30579577$ by $0$ .

$[\begin{array}{c} 5 & 5 & 5 & 5 \\ 6 & 5 & 5 & 6 \\ 7 & 8 & 9 & 8 \\ 6 & 7 & 8 & 9 \end{array}] + [\begin{array}{c} - 26.30579577 & 0 & 0 & 0 \\ 0 & - 26.30579577 & 0 & 0 \\ 0 & 0 & - 26.30579577 & 0 \\ 0 & 0 & 0 & - 26.30579577 \cdot 1 \end{array}]$

Step 4.2.1.2.16

Multiply $- 26.30579577$ by $1$ .

Step 4.2.2

Add the corresponding elements.

$[\begin{array}{c} 5 - 26.30579577 & 5 + 0 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 - 26.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3

Simplify each element.

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Step 4.2.3.1

Subtract $26.30579577$ from $5$ .

$[\begin{array}{c} - 21.30579577 & 5 + 0 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 - 26.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.2

Add $5$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 + 0 & 5 + 0 \\ 6 + 0 & 5 - 26.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.3

Add $5$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 + 0 \\ 6 + 0 & 5 - 26.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.4

Add $5$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 + 0 & 5 - 26.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.5

Add $6$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & 5 - 26.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.6

Subtract $26.30579577$ from $5$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 + 0 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.7

Add $5$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 + 0 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.8

Add $6$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 + 0 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.9

Add $7$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 + 0 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.10

Add $8$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & 9 - 26.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.11

Subtract $26.30579577$ from $9$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & - 17.30579577 & 8 + 0 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.12

Add $8$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & - 17.30579577 & 8 \\ 6 + 0 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.13

Add $6$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & - 17.30579577 & 8 \\ 6 & 7 + 0 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.14

Add $7$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & - 17.30579577 & 8 \\ 6 & 7 & 8 + 0 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.15

Add $8$ and $0$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & - 17.30579577 & 8 \\ 6 & 7 & 8 & 9 - 26.30579577 \end{array}]$

Step 4.2.3.16

Subtract $26.30579577$ from $9$ .

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 \\ 6 & - 21.30579577 & 5 & 6 \\ 7 & 8 & - 17.30579577 & 8 \\ 6 & 7 & 8 & - 17.30579577 \end{array}]$

Step 4.3

Find the null space when $λ = 26.30579577$ .

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Step 4.3.1

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

$[\begin{array}{c} - 21.30579577 & 5 & 5 & 5 & 0 \\ 6 & - 21.30579577 & 5 & 6 & 0 \\ 7 & 8 & - 17.30579577 & 8 & 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2

Find the reduced row echelon form.

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Step 4.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{- 21.30579577}$ to make the entry at $1, 1$ a $1$ .

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Step 4.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{- 21.30579577}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 21.30579577}{- 21.30579577} & \frac{5}{- 21.30579577} & \frac{5}{- 21.30579577} & \frac{5}{- 21.30579577} & \frac{0}{- 21.30579577} \\ 6 & - 21.30579577 & 5 & 6 & 0 \\ 7 & 8 & - 17.30579577 & 8 & 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 6 & - 21.30579577 & 5 & 6 & 0 \\ 7 & 8 & - 17.30579577 & 8 & 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 4.3.2.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 6 - 6 \cdot 1 & - 21.30579577 - 6 \cdot - 0.23467792 & 5 - 6 \cdot - 0.23467792 & 6 - 6 \cdot - 0.23467792 & 0 - 6 \cdot 0 \\ 7 & 8 & - 17.30579577 & 8 & 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & - 19.89772821 & 6.40806756 & 7.40806756 & 0 \\ 7 & 8 & - 17.30579577 & 8 & 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2.3

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 4.3.2.3.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & - 19.89772821 & 6.40806756 & 7.40806756 & 0 \\ 7 - 7 \cdot 1 & 8 - 7 \cdot - 0.23467792 & - 17.30579577 - 7 \cdot - 0.23467792 & 8 - 7 \cdot - 0.23467792 & 0 - 7 \cdot 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & - 19.89772821 & 6.40806756 & 7.40806756 & 0 \\ 0 & 9.64274549 & - 15.66305028 & 9.64274549 & 0 \\ 6 & 7 & 8 & - 17.30579577 & 0 \end{array}]$

Step 4.3.2.4

Perform the row operation $R_{4} = R_{4} - 6 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 4.3.2.4.1

Perform the row operation $R_{4} = R_{4} - 6 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & - 19.89772821 & 6.40806756 & 7.40806756 & 0 \\ 0 & 9.64274549 & - 15.66305028 & 9.64274549 & 0 \\ 6 - 6 \cdot 1 & 7 - 6 \cdot - 0.23467792 & 8 - 6 \cdot - 0.23467792 & - 17.30579577 - 6 \cdot - 0.23467792 & 0 - 6 \cdot 0 \end{array}]$

Step 4.3.2.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & - 19.89772821 & 6.40806756 & 7.40806756 & 0 \\ 0 & 9.64274549 & - 15.66305028 & 9.64274549 & 0 \\ 0 & 8.40806756 & 9.40806756 & - 15.89772821 & 0 \end{array}]$

Step 4.3.2.5

Multiply each element of $R_{2}$ by $\frac{1}{- 19.89772821}$ to make the entry at $2, 2$ a $1$ .

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Step 4.3.2.5.1

Multiply each element of $R_{2}$ by $\frac{1}{- 19.89772821}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ \frac{0}{- 19.89772821} & \frac{- 19.89772821}{- 19.89772821} & \frac{6.40806756}{- 19.89772821} & \frac{7.40806756}{- 19.89772821} & \frac{0}{- 19.89772821} \\ 0 & 9.64274549 & - 15.66305028 & 9.64274549 & 0 \\ 0 & 8.40806756 & 9.40806756 & - 15.89772821 & 0 \end{array}]$

Step 4.3.2.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 9.64274549 & - 15.66305028 & 9.64274549 & 0 \\ 0 & 8.40806756 & 9.40806756 & - 15.89772821 & 0 \end{array}]$

Step 4.3.2.6

Perform the row operation $R_{3} = R_{3} - 9.64274549 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.3.2.6.1

Perform the row operation $R_{3} = R_{3} - 9.64274549 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 - 9.64274549 \cdot 0 & 9.64274549 - 9.64274549 \cdot 1 & - 15.66305028 - 9.64274549 \cdot - 0.32205021 & 9.64274549 - 9.64274549 \cdot - 0.3723072 & 0 - 9.64274549 \cdot 0 \\ 0 & 8.40806756 & 9.40806756 & - 15.89772821 & 0 \end{array}]$

Step 4.3.2.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & - 12.55760206 & 13.23280911 & 0 \\ 0 & 8.40806756 & 9.40806756 & - 15.89772821 & 0 \end{array}]$

Step 4.3.2.7

Perform the row operation $R_{4} = R_{4} - 8.40806756 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 4.3.2.7.1

Perform the row operation $R_{4} = R_{4} - 8.40806756 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & - 12.55760206 & 13.23280911 & 0 \\ 0 - 8.40806756 \cdot 0 & 8.40806756 - 8.40806756 \cdot 1 & 9.40806756 - 8.40806756 \cdot - 0.32205021 & - 15.89772821 - 8.40806756 \cdot - 0.3723072 & 0 - 8.40806756 \cdot 0 \end{array}]$

Step 4.3.2.7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & - 12.55760206 & 13.23280911 & 0 \\ 0 & 0 & 12.11588749 & - 12.76734408 & 0 \end{array}]$

Step 4.3.2.8

Multiply each element of $R_{3}$ by $\frac{1}{- 12.55760206}$ to make the entry at $3, 3$ a $1$ .

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Step 4.3.2.8.1

Multiply each element of $R_{3}$ by $\frac{1}{- 12.55760206}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ \frac{0}{- 12.55760206} & \frac{0}{- 12.55760206} & \frac{- 12.55760206}{- 12.55760206} & \frac{13.23280911}{- 12.55760206} & \frac{0}{- 12.55760206} \\ 0 & 0 & 12.11588749 & - 12.76734408 & 0 \end{array}]$

Step 4.3.2.8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & 1 & - 1.05376878 & 0 \\ 0 & 0 & 12.11588749 & - 12.76734408 & 0 \end{array}]$

Step 4.3.2.9

Perform the row operation $R_{4} = R_{4} - 12.11588749 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 4.3.2.9.1

Perform the row operation $R_{4} = R_{4} - 12.11588749 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & 1 & - 1.05376878 & 0 \\ 0 - 12.11588749 \cdot 0 & 0 - 12.11588749 \cdot 0 & 12.11588749 - 12.11588749 \cdot 1 & - 12.76734408 - 12.11588749 \cdot - 1.05376878 & 0 - 12.11588749 \cdot 0 \end{array}]$

Step 4.3.2.9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & 1 & - 1.05376878 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.10

Multiply each element of $R_{4}$ by $\frac{1}{- 4.60593313 \cdot 10^{- 14}}$ to make the entry at $4, 4$ a $1$ .

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Step 4.3.2.10.1

Multiply each element of $R_{4}$ by $\frac{1}{- 4.60593313 \cdot 10^{- 14}}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & 1 & - 1.05376878 & 0 \\ \frac{0}{- 4.60593313 \cdot 10^{- 14}} & \frac{0}{- 4.60593313 \cdot 10^{- 14}} & \frac{0}{- 4.60593313 \cdot 10^{- 14}} & \frac{0}{- 4.60593313 \cdot 10^{- 14}} & \frac{0}{- 4.60593313 \cdot 10^{- 14}} \end{array}]$

Step 4.3.2.10.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & 1 & - 1.05376878 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.11

Perform the row operation $R_{3} = R_{3} + 1.05376878 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 4.3.2.11.1

Perform the row operation $R_{3} = R_{3} + 1.05376878 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 + 1.05376878 \cdot 0 & 0 + 1.05376878 \cdot 0 & 1 + 1.05376878 \cdot 0 & - 1.05376878 + 1.05376878 \cdot 1 & 0 + 1.05376878 \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.11.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & - 0.3723072 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.12

Perform the row operation $R_{2} = R_{2} + 0.3723072 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 4.3.2.12.1

Perform the row operation $R_{2} = R_{2} + 0.3723072 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 + 0.3723072 \cdot 0 & 1 + 0.3723072 \cdot 0 & - 0.32205021 + 0.3723072 \cdot 0 & - 0.3723072 + 0.3723072 \cdot 1 & 0 + 0.3723072 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.12.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & - 0.23467792 & 0 \\ 0 & 1 & - 0.32205021 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.13

Perform the row operation $R_{1} = R_{1} + 0.23467792 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Step 4.3.2.13.1

Perform the row operation $R_{1} = R_{1} + 0.23467792 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 0.23467792 \cdot 0 & - 0.23467792 + 0.23467792 \cdot 0 & - 0.23467792 + 0.23467792 \cdot 0 & - 0.23467792 + 0.23467792 \cdot 1 & 0 + 0.23467792 \cdot 0 \\ 0 & 1 & - 0.32205021 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.13.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & 0 & 0 \\ 0 & 1 & - 0.32205021 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.14

Perform the row operation $R_{2} = R_{2} + 0.32205021 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 4.3.2.14.1

Perform the row operation $R_{2} = R_{2} + 0.32205021 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & 0 & 0 \\ 0 + 0.32205021 \cdot 0 & 1 + 0.32205021 \cdot 0 & - 0.32205021 + 0.32205021 \cdot 1 & 0 + 0.32205021 \cdot 0 & 0 + 0.32205021 \cdot 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.14.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 0.23467792 & - 0.23467792 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.15

Perform the row operation $R_{1} = R_{1} + 0.23467792 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 4.3.2.15.1

Perform the row operation $R_{1} = R_{1} + 0.23467792 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 0.23467792 \cdot 0 & - 0.23467792 + 0.23467792 \cdot 0 & - 0.23467792 + 0.23467792 \cdot 1 & 0 + 0.23467792 \cdot 0 & 0 + 0.23467792 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 0.23467792 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.16

Perform the row operation $R_{1} = R_{1} + 0.23467792 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.3.2.16.1

Perform the row operation $R_{1} = R_{1} + 0.23467792 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 0.23467792 \cdot 0 & - 0.23467792 + 0.23467792 \cdot 1 & 0 + 0.23467792 \cdot 0 & 0 + 0.23467792 \cdot 0 & 0 + 0.23467792 \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 4.3.2.16.2

Simplify $R_{1}$ .

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

Step 4.3.3

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

$x_{1} = 0$

$x_{2} = 0$

$x_{3} = 0$

$x_{4} = 0$

Step 4.3.4

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

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

Step 4.3.5

Write as a solution set.

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

Step 5

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

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