Linear Algebra Examples

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 3 & 2 & 11 0 & - 1 & 3 & 8 0 & 0 & - 2 & 4 0 & 0 & 0 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}]$

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

$[\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}]$ for

$A$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] - λ I_{4})$

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

$I_{4}$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

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.

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.5.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.9.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.10

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.10.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.10.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.11

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.12

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.12.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.12.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.13

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.13.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.13.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.14

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.14.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.14.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.15

Multiply

$- λ \cdot 0$ .

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Step 4.1.2.15.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.15.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.16

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & 3 & 2 & 11 \\ 0 & - 1 & 3 & 8 \\ 0 & 0 & - 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 + 0 & 2 + 0 & 11 + 0 \\ 0 + 0 & - 1 - λ & 3 + 0 & 8 + 0 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add

$3$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 + 0 & 11 + 0 \\ 0 + 0 & - 1 - λ & 3 + 0 & 8 + 0 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.2

Add

$2$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 + 0 \\ 0 + 0 & - 1 - λ & 3 + 0 & 8 + 0 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.3

Add

$11$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 + 0 & - 1 - λ & 3 + 0 & 8 + 0 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.4

Add

$0$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 + 0 & 8 + 0 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.5

Add

$3$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 + 0 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.6

Add

$8$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 + 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.7

Add

$0$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 & 0 + 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.8

Add

$0$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 & 0 & - 2 - λ & 4 + 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.9

Add

$4$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 & 0 & - 2 - λ & 4 \\ 0 + 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.10

Add

$0$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 & 0 & - 2 - λ & 4 \\ 0 & 0 + 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.11

Add

$0$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 & 0 & - 2 - λ & 4 \\ 0 & 0 & 0 + 0 & 2 - λ \end{array}]$

Step 4.3.12

Add

$0$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & 3 & 2 & 11 \\ 0 & - 1 - λ & 3 & 8 \\ 0 & 0 & - 2 - λ & 4 \\ 0 & 0 & 0 & 2 - λ \end{array}]$

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.

$| \begin{array}{c} - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} |$

Step 5.1.4

Multiply element

$a_{11}$ by its cofactor.

$(1 - λ) | \begin{array}{c} - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} |$

Step 5.1.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 3 & 2 & 11 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} |$

Step 5.1.6

Multiply element

$a_{21}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 11 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} |$

Step 5.1.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

$| \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & 0 & 2 - λ \end{array} |$

Step 5.1.8

Multiply element

$a_{31}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & 0 & 2 - λ \end{array} |$

Step 5.1.9

The minor for

$a_{41}$ is the determinant with row

$4$ and column

$1$ deleted.

$| \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \end{array} |$

Step 5.1.10

Multiply element

$a_{41}$ by its cofactor.

$0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \end{array} |$

Step 5.1.11

Add the terms together.

$p (λ) = (1 - λ) | \begin{array}{c} - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} | + 0 | \begin{array}{c} 3 & 2 & 11 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} | + 0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & 0 & 2 - λ \end{array} | + 0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \end{array} |$

Step 5.2

Multiply

$0$ by

$| \begin{array}{c} 3 & 2 & 11 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} |$ .

$p (λ) = (1 - λ) | \begin{array}{c} - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} | + 0 + 0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & 0 & 2 - λ \end{array} | + 0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \end{array} |$

Step 5.3

Multiply

$0$ by

$| \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & 0 & 2 - λ \end{array} |$ .

$p (λ) = (1 - λ) | \begin{array}{c} - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} | + 0 + 0 + 0 | \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \end{array} |$

Step 5.4

Multiply

$0$ by

$| \begin{array}{c} 3 & 2 & 11 \\ - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \end{array} |$ .

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

Step 5.5

Evaluate

$| \begin{array}{c} - 1 - λ & 3 & 8 \\ 0 & - 2 - λ & 4 \\ 0 & 0 & 2 - λ \end{array} |$ .

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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

$1$ 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

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

Step 5.5.1.4

Multiply element

$a_{11}$ by its cofactor.

$(- 1 - λ) | \begin{array}{c} - 2 - λ & 4 \\ 0 & 2 - λ \end{array} |$

Step 5.5.1.5

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 3 & 8 \\ 0 & 2 - λ \end{array} |$

Step 5.5.1.6

Multiply element

$a_{21}$ by its cofactor.

$0 | \begin{array}{c} 3 & 8 \\ 0 & 2 - λ \end{array} |$

Step 5.5.1.7

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

$| \begin{array}{c} 3 & 8 \\ - 2 - λ & 4 \end{array} |$

Step 5.5.1.8

Multiply element

$a_{31}$ by its cofactor.

$0 | \begin{array}{c} 3 & 8 \\ - 2 - λ & 4 \end{array} |$

Step 5.5.1.9

Add the terms together.

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

Step 5.5.2

Multiply

$0$ by

$| \begin{array}{c} 3 & 8 \\ 0 & 2 - λ \end{array} |$ .

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

Step 5.5.3

Multiply

$0$ by

$| \begin{array}{c} 3 & 8 \\ - 2 - λ & 4 \end{array} |$ .

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

Step 5.5.4

Evaluate

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

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Step 5.5.4.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (1 - λ) ((- 1 - λ) ((- 2 - λ) (2 - λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

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

Expand

$(- 2 - λ) (2 - λ)$ using the FOIL Method.

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Step 5.5.4.2.1.1.1

Apply the distributive property.

$p (λ) = (1 - λ) ((- 1 - λ) (- 2 (2 - λ) - λ (2 - λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.1.2

Apply the distributive property.

$p (λ) = (1 - λ) ((- 1 - λ) (- 2 \cdot 2 - 2 (- λ) - λ (2 - λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.1.3

Apply the distributive property.

$p (λ) = (1 - λ) ((- 1 - λ) (- 2 \cdot 2 - 2 (- λ) - λ \cdot 2 - λ (- λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2

Simplify and combine like terms.

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Step 5.5.4.2.1.2.1

Simplify each term.

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Step 5.5.4.2.1.2.1.1

Multiply

$- 2$ by

$2$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 - 2 (- λ) - λ \cdot 2 - λ (- λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.2

Multiply

$- 1$ by

$- 2$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - λ \cdot 2 - λ (- λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.3

Multiply

$2$ by

$- 1$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - 2 λ - λ (- λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - 2 λ - 1 \cdot - 1 λ \cdot λ + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.5

Multiply

$λ$ by

$λ$ by adding the exponents.

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Step 5.5.4.2.1.2.1.5.1

Move

$λ$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - 2 λ - 1 \cdot - 1 (λ \cdot λ) + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.5.2

Multiply

$λ$ by

$λ$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - 2 λ - 1 \cdot - 1 λ^{2} + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.6

Multiply

$- 1$ by

$- 1$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - 2 λ + 1 λ^{2} + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.1.7

Multiply

$λ^{2}$ by

$1$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 2 λ - 2 λ + λ^{2} + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.2

Subtract

$2 λ$ from

$2 λ$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + 0 + λ^{2} + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.2.3

Add

$- 4$ and

$0$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + λ^{2} + 0 \cdot 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.1.3

Multiply

$0$ by

$4$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + λ^{2} + 0) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.2

Add

$- 4 + λ^{2}$ and

$0$ .

$p (λ) = (1 - λ) ((- 1 - λ) (- 4 + λ^{2}) + 0 + 0) + 0 + 0 + 0$

Step 5.5.4.2.3

Reorder

$- 4$ and

$λ^{2}$ .

$p (λ) = (1 - λ) ((- 1 - λ) (λ^{2} - 4) + 0 + 0) + 0 + 0 + 0$

Step 5.5.5

Simplify the determinant.

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Step 5.5.5.1

Combine the opposite terms in

$(- 1 - λ) (λ^{2} - 4) + 0 + 0$ .

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Step 5.5.5.1.1

Add

$(- 1 - λ) (λ^{2} - 4)$ and

$0$ .

$p (λ) = (1 - λ) ((- 1 - λ) (λ^{2} - 4) + 0) + 0 + 0 + 0$

Step 5.5.5.1.2

Add

$(- 1 - λ) (λ^{2} - 4)$ and

$0$ .

$p (λ) = (1 - λ) ((- 1 - λ) (λ^{2} - 4)) + 0 + 0 + 0$

Step 5.5.5.2

Expand

$(- 1 - λ) (λ^{2} - 4)$ using the FOIL Method.

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Step 5.5.5.2.1

Apply the distributive property.

$p (λ) = (1 - λ) (- 1 (λ^{2} - 4) - λ (λ^{2} - 4)) + 0 + 0 + 0$

Step 5.5.5.2.2

Apply the distributive property.

$p (λ) = (1 - λ) (- 1 λ^{2} - 1 \cdot - 4 - λ (λ^{2} - 4)) + 0 + 0 + 0$

Step 5.5.5.2.3

Apply the distributive property.

$p (λ) = (1 - λ) (- 1 λ^{2} - 1 \cdot - 4 - λ \cdot λ^{2} - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3

Simplify each term.

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Step 5.5.5.3.1

Rewrite

$- 1 λ^{2}$ as

$- λ^{2}$ .

$p (λ) = (1 - λ) (- λ^{2} - 1 \cdot - 4 - λ \cdot λ^{2} - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3.2

Multiply

$- 1$ by

$- 4$ .

$p (λ) = (1 - λ) (- λ^{2} + 4 - λ \cdot λ^{2} - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3.3

Multiply

$λ$ by

$λ^{2}$ by adding the exponents.

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Step 5.5.5.3.3.1

Move

$λ^{2}$ .

$p (λ) = (1 - λ) (- λ^{2} + 4 - (λ^{2} λ) - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3.3.2

Multiply

$λ^{2}$ by

$λ$ .

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Step 5.5.5.3.3.2.1

Raise

$λ$ to the power of

$1$ .

$p (λ) = (1 - λ) (- λ^{2} + 4 - (λ^{2} λ^{1}) - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3.3.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = (1 - λ) (- λ^{2} + 4 - λ^{2 + 1} - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3.3.3

Add

$2$ and

$1$ .

$p (λ) = (1 - λ) (- λ^{2} + 4 - λ^{3} - λ \cdot - 4) + 0 + 0 + 0$

Step 5.5.5.3.4

Multiply

$- 4$ by

$- 1$ .

$p (λ) = (1 - λ) (- λ^{2} + 4 - λ^{3} + 4 λ) + 0 + 0 + 0$

Step 5.5.5.4

Move

$4$ .

$p (λ) = (1 - λ) (- λ^{2} - λ^{3} + 4 λ + 4) + 0 + 0 + 0$

Step 5.5.5.5

Reorder

$- λ^{2}$ and

$- λ^{3}$ .

$p (λ) = (1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4) + 0 + 0 + 0$

Step 5.6

Simplify the determinant.

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Step 5.6.1

Combine the opposite terms in

$(1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4) + 0 + 0 + 0$ .

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Step 5.6.1.1

Add

$(1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4)$ and

$0$ .

$p (λ) = (1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4) + 0 + 0$

Step 5.6.1.2

Add

$(1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4)$ and

$0$ .

$p (λ) = (1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4) + 0$

Step 5.6.1.3

Add

$(1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4)$ and

$0$ .

$p (λ) = (1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4)$

Step 5.6.2

Expand

$(1 - λ) (- λ^{3} - λ^{2} + 4 λ + 4)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = 1 (- λ^{3}) + 1 (- λ^{2}) + 1 (4 λ) + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ) - λ \cdot 4$

Step 5.6.3

Combine the opposite terms in

$1 (- λ^{3}) + 1 (- λ^{2}) + 1 (4 λ) + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ) - λ \cdot 4$ .

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Step 5.6.3.1

Reorder the factors in the terms

$1 (4 λ)$ and

$- λ \cdot 4$ .

$p (λ) = 1 (- λ^{3}) + 1 (- λ^{2}) + 1 \cdot 4 λ + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ) - 1 \cdot 4 λ$

Step 5.6.3.2

Subtract

$4 λ$ from

$1 \cdot 4 λ$ .

$p (λ) = 1 (- λ^{3}) + 1 (- λ^{2}) + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ) + 0$

Step 5.6.3.3

Add

$1 (- λ^{3}) + 1 (- λ^{2}) + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ)$ and

$0$ .

$p (λ) = 1 (- λ^{3}) + 1 (- λ^{2}) + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4

Simplify each term.

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Step 5.6.4.1

Multiply

$- λ^{3}$ by

$1$ .

$p (λ) = - λ^{3} + 1 (- λ^{2}) + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.2

Multiply

$- λ^{2}$ by

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 1 \cdot 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.3

Multiply

$4$ by

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 - λ (- λ^{3}) - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.4

Rewrite using the commutative property of multiplication.

$p (λ) = - λ^{3} - λ^{2} + 4 - 1 \cdot - 1 λ \cdot λ^{3} - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.5

Multiply

$λ$ by

$λ^{3}$ by adding the exponents.

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Step 5.6.4.5.1

Move

$λ^{3}$ .

$p (λ) = - λ^{3} - λ^{2} + 4 - 1 \cdot - 1 (λ^{3} λ) - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.5.2

Multiply

$λ^{3}$ by

$λ$ .

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Step 5.6.4.5.2.1

Raise

$λ$ to the power of

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 - 1 \cdot - 1 (λ^{3} λ^{1}) - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.5.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - λ^{3} - λ^{2} + 4 - 1 \cdot - 1 λ^{3 + 1} - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.5.3

Add

$3$ and

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 - 1 \cdot - 1 λ^{4} - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.6

Multiply

$- 1$ by

$- 1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + 1 λ^{4} - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.7

Multiply

$λ^{4}$ by

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} - λ (- λ^{2}) - λ (4 λ)$

Step 5.6.4.8

Rewrite using the commutative property of multiplication.

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} - 1 \cdot - 1 λ \cdot λ^{2} - λ (4 λ)$

Step 5.6.4.9

Multiply

$λ$ by

$λ^{2}$ by adding the exponents.

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Step 5.6.4.9.1

Move

$λ^{2}$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} - 1 \cdot - 1 (λ^{2} λ) - λ (4 λ)$

Step 5.6.4.9.2

Multiply

$λ^{2}$ by

$λ$ .

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Step 5.6.4.9.2.1

Raise

$λ$ to the power of

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} - 1 \cdot - 1 (λ^{2} λ^{1}) - λ (4 λ)$

Step 5.6.4.9.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} - 1 \cdot - 1 λ^{2 + 1} - λ (4 λ)$

Step 5.6.4.9.3

Add

$2$ and

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} - 1 \cdot - 1 λ^{3} - λ (4 λ)$

Step 5.6.4.10

Multiply

$- 1$ by

$- 1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} + 1 λ^{3} - λ (4 λ)$

Step 5.6.4.11

Multiply

$λ^{3}$ by

$1$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} + λ^{3} - λ (4 λ)$

Step 5.6.4.12

Rewrite using the commutative property of multiplication.

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} + λ^{3} - 1 \cdot 4 λ \cdot λ$

Step 5.6.4.13

Multiply

$λ$ by

$λ$ by adding the exponents.

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Step 5.6.4.13.1

Move

$λ$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} + λ^{3} - 1 \cdot 4 (λ \cdot λ)$

Step 5.6.4.13.2

Multiply

$λ$ by

$λ$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} + λ^{3} - 1 \cdot 4 λ^{2}$

Step 5.6.4.14

Multiply

$- 1$ by

$4$ .

$p (λ) = - λ^{3} - λ^{2} + 4 + λ^{4} + λ^{3} - 4 λ^{2}$

Step 5.6.5

Combine the opposite terms in

$- λ^{3} - λ^{2} + 4 + λ^{4} + λ^{3} - 4 λ^{2}$ .

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Step 5.6.5.1

Add

$- λ^{3}$ and

$λ^{3}$ .

$p (λ) = - λ^{2} + 4 + λ^{4} + 0 - 4 λ^{2}$

Step 5.6.5.2

Add

$- λ^{2} + 4 + λ^{4}$ and

$0$ .

$p (λ) = - λ^{2} + 4 + λ^{4} - 4 λ^{2}$

Step 5.6.6

Subtract

$4 λ^{2}$ from

$- λ^{2}$ .

$p (λ) = - 5 λ^{2} + 4 + λ^{4}$

Step 5.6.7

Move

$4$ .

$p (λ) = - 5 λ^{2} + λ^{4} + 4$

Step 5.6.8

Reorder

$- 5 λ^{2}$ and

$λ^{4}$ .

$p (λ) = λ^{4} - 5 λ^{2} + 4$

Step 6

Set the characteristic polynomial equal to

$0$ to find the eigenvalues

$λ$ .

$λ^{4} - 5 λ^{2} + 4 = 0$

Step 7

Solve for

$λ$ .

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Step 7.1

Substitute

$u = λ^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 5 u + 4 = 0$

$u = λ^{2}$

Step 7.2

Factor

$u^{2} - 5 u + 4$ using the AC method.

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Step 7.2.1

Consider the form

$x^{2} + b x + c$ . Find a pair of integers whose product is

$c$ and whose sum is

$b$ . In this case, whose product is

$4$ and whose sum is

$- 5$ .

$- 4, - 1$

Step 7.2.2

Write the factored form using these integers.

$(u - 4) (u - 1) = 0$

Step 7.3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$u - 4 = 0$

$u - 1 = 0$

Step 7.4

Set

$u - 4$ equal to

$0$ and solve for

$u$ .

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Step 7.4.1

Set

$u - 4$ equal to

$0$ .

$u - 4 = 0$

Step 7.4.2

Add

$4$ to both sides of the equation.

$u = 4$

Step 7.5

Set

$u - 1$ equal to

$0$ and solve for

$u$ .

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Step 7.5.1

Set

$u - 1$ equal to

$0$ .

$u - 1 = 0$

Step 7.5.2

Add

$1$ to both sides of the equation.

$u = 1$

Step 7.6

The final solution is all the values that make

$(u - 4) (u - 1) = 0$ true.

$u = 4, 1$

Step 7.7

Substitute the real value of

$u = λ^{2}$ back into the solved equation.

$λ^{2} = 4$

${(λ^{2})}^{1} = 1$

Step 7.8

Solve the first equation for

$λ$ .

$λ^{2} = 4$

Step 7.9

Solve the equation for

$λ$ .

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Step 7.9.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$λ = \pm \sqrt{4}$

Step 7.9.2

Simplify

$\pm \sqrt{4}$ .

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Step 7.9.2.1

Rewrite

$4$ as

$2^{2}$ .

$λ = \pm \sqrt{2^{2}}$

Step 7.9.2.2

Pull terms out from under the radical, assuming positive real numbers.

$λ = \pm 2$

Step 7.9.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.9.3.1

First, use the positive value of the

$\pm$ to find the first solution.

$λ = 2$

Step 7.9.3.2

Next, use the negative value of the

$\pm$ to find the second solution.

$λ = - 2$

Step 7.9.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$λ = 2, - 2$

Step 7.10

Solve the second equation for

$λ$ .

${(λ^{2})}^{1} = 1$

Step 7.11

Solve the equation for

$λ$ .

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Step 7.11.1

Remove parentheses.

$λ^{2} = 1$

Step 7.11.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$λ = \pm \sqrt{1}$

Step 7.11.3

Any root of

$1$ is

$1$ .

$λ = \pm 1$

Step 7.11.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.11.4.1

First, use the positive value of the

$\pm$ to find the first solution.

$λ = 1$

Step 7.11.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

$λ = - 1$

Step 7.11.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$λ = 1, - 1$

Step 7.12

The solution to

$λ^{4} - 5 λ^{2} + 4 = 0$ is

$λ = 2, - 2, 1, - 1$ .

$λ = 2, - 2, 1, - 1$