Linear Algebra Examples

$[\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}]$

Step 1

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

$p (λ) = determinant (A - λ I_{4})$

Step 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 3

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

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Step 3.1

Substitute $[\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] - λ I_{4})$

Step 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 $I_{4}$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply $- λ \cdot 0$ .

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Step 4.1.2.5.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply $- λ \cdot 0$ .

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Step 4.1.2.9.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.10

Multiply $- λ \cdot 0$ .

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Step 4.1.2.10.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.10.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.11

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.12

Multiply $- λ \cdot 0$ .

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Step 4.1.2.12.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.12.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.13

Multiply $- λ \cdot 0$ .

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Step 4.1.2.13.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.13.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.14

Multiply $- λ \cdot 0$ .

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Step 4.1.2.14.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.14.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.15

Multiply $- λ \cdot 0$ .

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Step 4.1.2.15.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.15.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.16

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 & 0 & 13 \\ 2 & 0 & - 12 & - 8 \\ 0 & 4 & 0 & - 8 \\ 0 & 0 & 8 & 8 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 + 0 & 0 + 0 & 13 + 0 \\ 2 + 0 & 0 - λ & - 12 + 0 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add $- 2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 + 0 & 13 + 0 \\ 2 + 0 & 0 - λ & - 12 + 0 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.2

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 + 0 \\ 2 + 0 & 0 - λ & - 12 + 0 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.3

Add $13$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 + 0 & 0 - λ & - 12 + 0 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.4

Add $2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & 0 - λ & - 12 + 0 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.5

Subtract $λ$ from $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 + 0 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.6

Add $- 12$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 + 0 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.7

Add $- 8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 + 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.8

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 + 0 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.9

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 & 0 - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.10

Subtract $λ$ from $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 & - λ & - 8 + 0 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.11

Add $- 8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 & - λ & - 8 \\ 0 + 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.12

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 & - λ & - 8 \\ 0 & 0 + 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.13

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 & - λ & - 8 \\ 0 & 0 & 8 + 0 & 8 - λ \end{array}]$

Step 4.3.14

Add $8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 & 0 & 13 \\ 2 & - λ & - 12 & - 8 \\ 0 & 4 & - λ & - 8 \\ 0 & 0 & 8 & 8 - λ \end{array}]$

Step 5

Find the determinant.

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Step 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 column $1$ by its cofactor and add.

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Step 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

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

Step 5.1.3

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

$| \begin{array}{c} - λ & - 12 & - 8 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} |$

Step 5.1.4

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

$(1 - λ) | \begin{array}{c} - λ & - 12 & - 8 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} |$

Step 5.1.6

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

$- 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} |$

Step 5.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 0 & 8 & 8 - λ \end{array} |$

Step 5.1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 0 & 8 & 8 - λ \end{array} |$

Step 5.1.9

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 4 & - λ & - 8 \end{array} |$

Step 5.1.10

Multiply element $a_{41}$ by its cofactor.

$0 | \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 4 & - λ & - 8 \end{array} |$

Step 5.1.11

Add the terms together.

$p (λ) = (1 - λ) | \begin{array}{c} - λ & - 12 & - 8 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 | \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 | \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 4 & - λ & - 8 \end{array} |$

Step 5.2

Multiply $0$ by $| \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 0 & 8 & 8 - λ \end{array} |$ .

$p (λ) = (1 - λ) | \begin{array}{c} - λ & - 12 & - 8 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0 | \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 4 & - λ & - 8 \end{array} |$

Step 5.3

Multiply $0$ by $| \begin{array}{c} - 2 & 0 & 13 \\ - λ & - 12 & - 8 \\ 4 & - λ & - 8 \end{array} |$ .

$p (λ) = (1 - λ) | \begin{array}{c} - λ & - 12 & - 8 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4

Evaluate $| \begin{array}{c} - λ & - 12 & - 8 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} |$ .

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Step 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 column $1$ by its cofactor and add.

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Step 5.4.1.1

Consider the corresponding sign chart.

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

Step 5.4.1.2

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

Step 5.4.1.3

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

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

Step 5.4.1.4

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

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

Step 5.4.1.5

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

$| \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} |$

Step 5.4.1.6

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

$- 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} |$

Step 5.4.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} - 12 & - 8 \\ - λ & - 8 \end{array} |$

Step 5.4.1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} - 12 & - 8 \\ - λ & - 8 \end{array} |$

Step 5.4.1.9

Add the terms together.

$p (λ) = (1 - λ) (- λ | \begin{array}{c} - λ & - 8 \\ 8 & 8 - λ \end{array} | - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0 | \begin{array}{c} - 12 & - 8 \\ - λ & - 8 \end{array} |) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.2

Multiply $0$ by $| \begin{array}{c} - 12 & - 8 \\ - λ & - 8 \end{array} |$ .

$p (λ) = (1 - λ) (- λ | \begin{array}{c} - λ & - 8 \\ 8 & 8 - λ \end{array} | - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3

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

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Step 5.4.3.1

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

$p (λ) = (1 - λ) (- λ (- λ (8 - λ) - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2

Simplify the determinant.

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Step 5.4.3.2.1

Simplify each term.

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Step 5.4.3.2.1.1

Apply the distributive property.

$p (λ) = (1 - λ) (- λ (- λ \cdot 8 - λ (- λ) - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.2

Multiply $8$ by $- 1$ .

$p (λ) = (1 - λ) (- λ (- 8 λ - λ (- λ) - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.3

Rewrite using the commutative property of multiplication.

$p (λ) = (1 - λ) (- λ (- 8 λ - 1 \cdot - 1 λ \cdot λ - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.4

Simplify each term.

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Step 5.4.3.2.1.4.1

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

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Step 5.4.3.2.1.4.1.1

Move $λ$ .

$p (λ) = (1 - λ) (- λ (- 8 λ - 1 \cdot - 1 (λ \cdot λ) - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.4.1.2

Multiply $λ$ by $λ$ .

$p (λ) = (1 - λ) (- λ (- 8 λ - 1 \cdot - 1 λ^{2} - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.4.2

Multiply $- 1$ by $- 1$ .

$p (λ) = (1 - λ) (- λ (- 8 λ + 1 λ^{2} - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.4.3

Multiply $λ^{2}$ by $1$ .

$p (λ) = (1 - λ) (- λ (- 8 λ + λ^{2} - 8 \cdot - 8) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.1.5

Multiply $- 8$ by $- 8$ .

$p (λ) = (1 - λ) (- λ (- 8 λ + λ^{2} + 64) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.3.2.2

Reorder $- 8 λ$ and $λ^{2}$ .

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 | \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} | + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.4

Evaluate $| \begin{array}{c} - 12 & - 8 \\ 8 & 8 - λ \end{array} |$ .

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Step 5.4.4.1

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

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (- 12 (8 - λ) - 8 \cdot - 8) + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.4.2

Simplify the determinant.

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Step 5.4.4.2.1

Simplify each term.

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Step 5.4.4.2.1.1

Apply the distributive property.

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (- 12 \cdot 8 - 12 (- λ) - 8 \cdot - 8) + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.4.2.1.2

Multiply $- 12$ by $8$ .

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (- 96 - 12 (- λ) - 8 \cdot - 8) + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.4.2.1.3

Multiply $- 1$ by $- 12$ .

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (- 96 + 12 λ - 8 \cdot - 8) + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.4.2.1.4

Multiply $- 8$ by $- 8$ .

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (- 96 + 12 λ + 64) + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.4.2.2

Add $- 96$ and $64$ .

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (12 λ - 32) + 0) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5

Simplify the determinant.

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Step 5.4.5.1

Add $- λ (λ^{2} - 8 λ + 64) - 4 (12 λ - 32)$ and $0$ .

$p (λ) = (1 - λ) (- λ (λ^{2} - 8 λ + 64) - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2

Simplify each term.

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Step 5.4.5.2.1

Apply the distributive property.

$p (λ) = (1 - λ) (- λ \cdot λ^{2} - λ (- 8 λ) - λ \cdot 64 - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.2

Simplify.

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Step 5.4.5.2.2.1

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

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Step 5.4.5.2.2.1.1

Move $λ^{2}$ .

$p (λ) = (1 - λ) (- (λ^{2} λ) - λ (- 8 λ) - λ \cdot 64 - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.2.1.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.4.5.2.2.1.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = (1 - λ) (- (λ^{2} λ^{1}) - λ (- 8 λ) - λ \cdot 64 - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.2.1.2.2

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

$p (λ) = (1 - λ) (- λ^{2 + 1} - λ (- 8 λ) - λ \cdot 64 - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.2.1.3

Add $2$ and $1$ .

$p (λ) = (1 - λ) (- λ^{3} - λ (- 8 λ) - λ \cdot 64 - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.2.2

Rewrite using the commutative property of multiplication.

$p (λ) = (1 - λ) (- λ^{3} - 1 \cdot - 8 λ \cdot λ - λ \cdot 64 - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.2.3

Multiply $64$ by $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} - 1 \cdot - 8 λ \cdot λ - 64 λ - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.3

Simplify each term.

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Step 5.4.5.2.3.1

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

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Step 5.4.5.2.3.1.1

Move $λ$ .

$p (λ) = (1 - λ) (- λ^{3} - 1 \cdot - 8 (λ \cdot λ) - 64 λ - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.3.1.2

Multiply $λ$ by $λ$ .

$p (λ) = (1 - λ) (- λ^{3} - 1 \cdot - 8 λ^{2} - 64 λ - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.3.2

Multiply $- 1$ by $- 8$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 64 λ - 4 (12 λ - 32)) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.4

Apply the distributive property.

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 64 λ - 4 (12 λ) - 4 \cdot - 32) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.5

Multiply $12$ by $- 4$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 64 λ - 48 λ - 4 \cdot - 32) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.2.6

Multiply $- 4$ by $- 32$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 64 λ - 48 λ + 128) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.4.5.3

Subtract $48 λ$ from $- 64 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 | \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} | + 0 + 0$

Step 5.5

Evaluate $| \begin{array}{c} - 2 & 0 & 13 \\ 4 & - λ & - 8 \\ 0 & 8 & 8 - λ \end{array} |$ .

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Step 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 column $2$ by its cofactor and add.

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Step 5.5.1.1

Consider the corresponding sign chart.

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

Step 5.5.1.2

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

Step 5.5.1.3

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

$| \begin{array}{c} 4 & - 8 \\ 0 & 8 - λ \end{array} |$

Step 5.5.1.4

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

$0 | \begin{array}{c} 4 & - 8 \\ 0 & 8 - λ \end{array} |$

Step 5.5.1.5

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

$| \begin{array}{c} - 2 & 13 \\ 0 & 8 - λ \end{array} |$

Step 5.5.1.6

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

$- λ | \begin{array}{c} - 2 & 13 \\ 0 & 8 - λ \end{array} |$

Step 5.5.1.7

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |$

Step 5.5.1.8

Multiply element $a_{32}$ by its cofactor.

$- 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |$

Step 5.5.1.9

Add the terms together.

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 | \begin{array}{c} 4 & - 8 \\ 0 & 8 - λ \end{array} | - λ | \begin{array}{c} - 2 & 13 \\ 0 & 8 - λ \end{array} | - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.2

Multiply $0$ by $| \begin{array}{c} 4 & - 8 \\ 0 & 8 - λ \end{array} |$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ | \begin{array}{c} - 2 & 13 \\ 0 & 8 - λ \end{array} | - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3

Evaluate $| \begin{array}{c} - 2 & 13 \\ 0 & 8 - λ \end{array} |$ .

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Step 5.5.3.1

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

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (- 2 (8 - λ) + 0 \cdot 13) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3.2

Simplify the determinant.

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Step 5.5.3.2.1

Simplify each term.

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Step 5.5.3.2.1.1

Apply the distributive property.

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (- 2 \cdot 8 - 2 (- λ) + 0 \cdot 13) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3.2.1.2

Multiply $- 2$ by $8$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (- 16 - 2 (- λ) + 0 \cdot 13) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3.2.1.3

Multiply $- 1$ by $- 2$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (- 16 + 2 λ + 0 \cdot 13) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3.2.1.4

Multiply $0$ by $13$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (- 16 + 2 λ + 0) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3.2.2

Add $- 16 + 2 λ$ and $0$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (- 16 + 2 λ) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.3.2.3

Reorder $- 16$ and $2 λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (2 λ - 16) - 8 | \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |) + 0 + 0$

Step 5.5.4

Evaluate $| \begin{array}{c} - 2 & 13 \\ 4 & - 8 \end{array} |$ .

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Step 5.5.4.1

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

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (2 λ - 16) - 8 (- 2 \cdot - 8 - 4 \cdot 13)) + 0 + 0$

Step 5.5.4.2

Simplify the determinant.

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Step 5.5.4.2.1

Simplify each term.

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Step 5.5.4.2.1.1

Multiply $- 2$ by $- 8$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (2 λ - 16) - 8 (16 - 4 \cdot 13)) + 0 + 0$

Step 5.5.4.2.1.2

Multiply $- 4$ by $13$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (2 λ - 16) - 8 (16 - 52)) + 0 + 0$

Step 5.5.4.2.2

Subtract $52$ from $16$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (0 - λ (2 λ - 16) - 8 \cdot - 36) + 0 + 0$

Step 5.5.5

Simplify the determinant.

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Step 5.5.5.1

Subtract $λ (2 λ - 16)$ from $0$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- λ (2 λ - 16) - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2

Simplify each term.

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Step 5.5.5.2.1

Apply the distributive property.

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- λ (2 λ) - λ \cdot - 16 - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2.2

Rewrite using the commutative property of multiplication.

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 1 \cdot 2 λ \cdot λ - λ \cdot - 16 - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2.3

Multiply $- 16$ by $- 1$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 1 \cdot 2 λ \cdot λ + 16 λ - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2.4

Simplify each term.

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Step 5.5.5.2.4.1

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

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Step 5.5.5.2.4.1.1

Move $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 1 \cdot 2 (λ \cdot λ) + 16 λ - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2.4.1.2

Multiply $λ$ by $λ$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 1 \cdot 2 λ^{2} + 16 λ - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2.4.2

Multiply $- 1$ by $2$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ - 8 \cdot - 36) + 0 + 0$

Step 5.5.5.2.5

Multiply $- 8$ by $- 36$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ + 288) + 0 + 0$

Step 5.6

Simplify the determinant.

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Step 5.6.1

Combine the opposite terms in $(1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ + 288) + 0 + 0$ .

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Step 5.6.1.1

Add $(1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ + 288)$ and $0$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ + 288) + 0$

Step 5.6.1.2

Add $(1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ + 288)$ and $0$ .

$p (λ) = (1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128) - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2

Simplify each term.

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Step 5.6.2.1

Expand $(1 - λ) (- λ^{3} + 8 λ^{2} - 112 λ + 128)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = 1 (- λ^{3}) + 1 (8 λ^{2}) + 1 (- 112 λ) + 1 \cdot 128 - λ (- λ^{3}) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2

Simplify each term.

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Step 5.6.2.2.1

Multiply $- λ^{3}$ by $1$ .

$p (λ) = - λ^{3} + 1 (8 λ^{2}) + 1 (- 112 λ) + 1 \cdot 128 - λ (- λ^{3}) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.2

Multiply $8 λ^{2}$ by $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} + 1 (- 112 λ) + 1 \cdot 128 - λ (- λ^{3}) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.3

Multiply $- 112 λ$ by $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 1 \cdot 128 - λ (- λ^{3}) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.4

Multiply $128$ by $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 - λ (- λ^{3}) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.5

Rewrite using the commutative property of multiplication.

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 - 1 \cdot - 1 λ \cdot λ^{3} - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.6

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

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Step 5.6.2.2.6.1

Move $λ^{3}$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 - 1 \cdot - 1 (λ^{3} λ) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.6.2

Multiply $λ^{3}$ by $λ$ .

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Step 5.6.2.2.6.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 - 1 \cdot - 1 (λ^{3} λ^{1}) - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.6.2.2

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

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 - 1 \cdot - 1 λ^{3 + 1} - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.6.3

Add $3$ and $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 - 1 \cdot - 1 λ^{4} - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.7

Multiply $- 1$ by $- 1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + 1 λ^{4} - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.8

Multiply $λ^{4}$ by $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - λ (8 λ^{2}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.9

Rewrite using the commutative property of multiplication.

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 1 \cdot 8 λ \cdot λ^{2} - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.10

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

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Step 5.6.2.2.10.1

Move $λ^{2}$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 1 \cdot 8 (λ^{2} λ) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.10.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.6.2.2.10.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 1 \cdot 8 (λ^{2} λ^{1}) - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.10.2.2

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

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 1 \cdot 8 λ^{2 + 1} - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.10.3

Add $2$ and $1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 1 \cdot 8 λ^{3} - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.11

Multiply $- 1$ by $8$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 8 λ^{3} - λ (- 112 λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.12

Rewrite using the commutative property of multiplication.

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 8 λ^{3} - 1 \cdot - 112 λ \cdot λ - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.13

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

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Step 5.6.2.2.13.1

Move $λ$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 8 λ^{3} - 1 \cdot - 112 (λ \cdot λ) - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.13.2

Multiply $λ$ by $λ$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 8 λ^{3} - 1 \cdot - 112 λ^{2} - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.14

Multiply $- 1$ by $- 112$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 8 λ^{3} + 112 λ^{2} - λ \cdot 128 - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.2.15

Multiply $128$ by $- 1$ .

$p (λ) = - λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} - 8 λ^{3} + 112 λ^{2} - 128 λ - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.3

Subtract $8 λ^{3}$ from $- λ^{3}$ .

$p (λ) = - 9 λ^{3} + 8 λ^{2} - 112 λ + 128 + λ^{4} + 112 λ^{2} - 128 λ - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.4

Add $8 λ^{2}$ and $112 λ^{2}$ .

$p (λ) = - 9 λ^{3} + 120 λ^{2} - 112 λ + 128 + λ^{4} - 128 λ - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.5

Subtract $128 λ$ from $- 112 λ$ .

$p (λ) = - 9 λ^{3} + 120 λ^{2} - 240 λ + 128 + λ^{4} - 2 (- 2 λ^{2} + 16 λ + 288)$

Step 5.6.2.6

Apply the distributive property.

$p (λ) = - 9 λ^{3} + 120 λ^{2} - 240 λ + 128 + λ^{4} - 2 (- 2 λ^{2}) - 2 (16 λ) - 2 \cdot 288$

Step 5.6.2.7

Simplify.

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Step 5.6.2.7.1

Multiply $- 2$ by $- 2$ .

$p (λ) = - 9 λ^{3} + 120 λ^{2} - 240 λ + 128 + λ^{4} + 4 λ^{2} - 2 (16 λ) - 2 \cdot 288$

Step 5.6.2.7.2

Multiply $16$ by $- 2$ .

$p (λ) = - 9 λ^{3} + 120 λ^{2} - 240 λ + 128 + λ^{4} + 4 λ^{2} - 32 λ - 2 \cdot 288$

Step 5.6.2.7.3

Multiply $- 2$ by $288$ .

$p (λ) = - 9 λ^{3} + 120 λ^{2} - 240 λ + 128 + λ^{4} + 4 λ^{2} - 32 λ - 576$

Step 5.6.3

Add $120 λ^{2}$ and $4 λ^{2}$ .

$p (λ) = - 9 λ^{3} + 124 λ^{2} - 240 λ + 128 + λ^{4} - 32 λ - 576$

Step 5.6.4

Subtract $32 λ$ from $- 240 λ$ .

$p (λ) = - 9 λ^{3} + 124 λ^{2} - 272 λ + 128 + λ^{4} - 576$

Step 5.6.5

Subtract $576$ from $128$ .

$p (λ) = - 9 λ^{3} + 124 λ^{2} - 272 λ + λ^{4} - 448$

Step 5.6.6

Move $- 272 λ$ .

$p (λ) = - 9 λ^{3} + 124 λ^{2} + λ^{4} - 272 λ - 448$

Step 5.6.7

Move $124 λ^{2}$ .

$p (λ) = - 9 λ^{3} + λ^{4} + 124 λ^{2} - 272 λ - 448$

Step 5.6.8

Reorder $- 9 λ^{3}$ and $λ^{4}$ .

$p (λ) = λ^{4} - 9 λ^{3} + 124 λ^{2} - 272 λ - 448$

Step 6

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

$λ^{4} - 9 λ^{3} + 124 λ^{2} - 272 λ - 448 = 0$

Step 7

Solve for $λ$ .

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Step 7.1

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

$λ \approx - 1.07462826, 3.7522584$