Linear Algebra Examples

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

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

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

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 $I_{4}$ .

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

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.

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

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

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

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.5

Multiply $- λ \cdot 0$ .

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Step 4.1.2.5.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.5.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.6

Multiply $- 1$ by $1$ .

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

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.9

Multiply $- λ \cdot 0$ .

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Step 4.1.2.9.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.9.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.10

Multiply $- λ \cdot 0$ .

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Step 4.1.2.10.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.10.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.11

Multiply $- 1$ by $1$ .

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

Step 4.1.2.12

Multiply $- λ \cdot 0$ .

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Step 4.1.2.12.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.12.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.13

Multiply $- λ \cdot 0$ .

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Step 4.1.2.13.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.13.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.14

Multiply $- λ \cdot 0$ .

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Step 4.1.2.14.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.14.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.15

Multiply $- λ \cdot 0$ .

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Step 4.1.2.15.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.15.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.16

Multiply $- 1$ by $1$ .

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

Step 4.2

Add the corresponding elements.

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

Step 4.3

Simplify each element.

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Step 4.3.1

Add $0$ and $0$ .

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

Step 4.3.2

Add $0$ and $0$ .

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

Step 4.3.3

Add $0$ and $0$ .

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

Step 4.3.4

Add $0$ and $0$ .

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

Step 4.3.5

Add $0$ and $0$ .

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

Step 4.3.6

Add $0$ and $0$ .

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

Step 4.3.7

Add $0$ and $0$ .

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

Step 4.3.8

Add $0$ and $0$ .

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

Step 4.3.9

Add $0$ and $0$ .

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

Step 4.3.10

Add $0$ and $0$ .

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

Step 4.3.11

Add $0$ and $0$ .

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

Step 4.3.12

Add $0$ and $0$ .

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

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 row $1$ 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.

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

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

Step 5.1.9

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

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

Step 5.1.10

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

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

Step 5.1.11

Add the terms together.

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

Step 5.2

Multiply $0$ by $| \begin{array}{c} 0 & 0 & 0 \\ 0 & 1 - λ & 0 \\ 0 & 0 & 1 - λ \end{array} |$ .

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

Step 5.3

Multiply $0$ by $| \begin{array}{c} 0 & 1 - λ & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 - λ \end{array} |$ .

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

Step 5.4

Multiply $0$ by $| \begin{array}{c} 0 & 1 - λ & 0 \\ 0 & 0 & 1 - λ \\ 0 & 0 & 0 \end{array} |$ .

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

Step 5.5

Evaluate $| \begin{array}{c} 1 - λ & 0 & 0 \\ 0 & 1 - λ & 0 \\ 0 & 0 & 1 - λ \end{array} |$ .

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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 row $1$ 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.

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

Step 5.5.1.4

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

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

Step 5.5.1.5

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

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

Step 5.5.1.6

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

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

Step 5.5.1.7

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

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

Step 5.5.1.8

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

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

Step 5.5.1.9

Add the terms together.

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

Step 5.5.2

Multiply $0$ by $| \begin{array}{c} 0 & 0 \\ 0 & 1 - λ \end{array} |$ .

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

Step 5.5.3

Multiply $0$ by $| \begin{array}{c} 0 & 1 - λ \\ 0 & 0 \end{array} |$ .

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

Step 5.5.4

Evaluate $| \begin{array}{c} 1 - λ & 0 \\ 0 & 1 - λ \end{array} |$ .

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Step 5.5.4.1

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

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

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 $(1 - λ) (1 - λ)$ using the FOIL Method.

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Step 5.5.4.2.1.1.1

Apply the distributive property.

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

Step 5.5.4.2.1.1.2

Apply the distributive property.

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

Step 5.5.4.2.1.1.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 5.5.4.2.1.2.1.2

Multiply $- λ$ by $1$ .

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

Step 5.5.4.2.1.2.1.3

Multiply $- 1$ by $1$ .

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

Step 5.5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

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 $λ$ .

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

Step 5.5.4.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

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

Step 5.5.4.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

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

Step 5.5.4.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

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

Step 5.5.4.2.1.2.2

Subtract $λ$ from $- λ$ .

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

Step 5.5.4.2.1.3

Multiply $0$ by $0$ .

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

Step 5.5.4.2.2

Add $1 - 2 λ + λ^{2}$ and $0$ .

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

Step 5.5.4.2.3

Move $1$ .

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

Step 5.5.4.2.4

Reorder $- 2 λ$ and $λ^{2}$ .

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

Step 5.5.5

Simplify the determinant.

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Step 5.5.5.1

Combine the opposite terms in $(1 - λ) (λ^{2} - 2 λ + 1) + 0 + 0$ .

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Step 5.5.5.1.1

Add $(1 - λ) (λ^{2} - 2 λ + 1)$ and $0$ .

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

Step 5.5.5.1.2

Add $(1 - λ) (λ^{2} - 2 λ + 1)$ and $0$ .

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

Step 5.5.5.2

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

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

Step 5.5.5.3

Simplify each term.

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Step 5.5.5.3.1

Multiply $λ^{2}$ by $1$ .

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

Step 5.5.5.3.2

Multiply $- 2 λ$ by $1$ .

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

Step 5.5.5.3.3

Multiply $1$ by $1$ .

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

Step 5.5.5.3.4

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

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Step 5.5.5.3.4.1

Move $λ^{2}$ .

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

Step 5.5.5.3.4.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.5.5.3.4.2.1

Raise $λ$ to the power of $1$ .

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

Step 5.5.5.3.4.2.2

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

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

Step 5.5.5.3.4.3

Add $2$ and $1$ .

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

Step 5.5.5.3.5

Rewrite using the commutative property of multiplication.

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

Step 5.5.5.3.6

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

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Step 5.5.5.3.6.1

Move $λ$ .

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

Step 5.5.5.3.6.2

Multiply $λ$ by $λ$ .

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

Step 5.5.5.3.7

Multiply $- 1$ by $- 2$ .

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

Step 5.5.5.3.8

Multiply $- 1$ by $1$ .

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

Step 5.5.5.4

Add $λ^{2}$ and $2 λ^{2}$ .

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

Step 5.5.5.5

Subtract $λ$ from $- 2 λ$ .

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

Step 5.5.5.6

Move $1$ .

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

Step 5.5.5.7

Move $- 3 λ$ .

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

Step 5.5.5.8

Reorder $3 λ^{2}$ and $- λ^{3}$ .

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

Step 5.6

Simplify the determinant.

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Step 5.6.1

Combine the opposite terms in $(1 - λ) (- λ^{3} + 3 λ^{2} - 3 λ + 1) + 0 + 0 + 0$ .

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Step 5.6.1.1

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

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

Step 5.6.1.2

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

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

Step 5.6.1.3

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

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

Step 5.6.2

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

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

Step 5.6.3

Simplify each term.

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Step 5.6.3.1

Multiply $- λ^{3}$ by $1$ .

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

Step 5.6.3.2

Multiply $3 λ^{2}$ by $1$ .

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

Step 5.6.3.3

Multiply $- 3 λ$ by $1$ .

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

Step 5.6.3.4

Multiply $1$ by $1$ .

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

Step 5.6.3.5

Rewrite using the commutative property of multiplication.

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

Step 5.6.3.6

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

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Step 5.6.3.6.1

Move $λ^{3}$ .

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

Step 5.6.3.6.2

Multiply $λ^{3}$ by $λ$ .

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Step 5.6.3.6.2.1

Raise $λ$ to the power of $1$ .

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

Step 5.6.3.6.2.2

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

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

Step 5.6.3.6.3

Add $3$ and $1$ .

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

Step 5.6.3.7

Multiply $- 1$ by $- 1$ .

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

Step 5.6.3.8

Multiply $λ^{4}$ by $1$ .

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

Step 5.6.3.9

Rewrite using the commutative property of multiplication.

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

Step 5.6.3.10

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

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Step 5.6.3.10.1

Move $λ^{2}$ .

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

Step 5.6.3.10.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.6.3.10.2.1

Raise $λ$ to the power of $1$ .

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

Step 5.6.3.10.2.2

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

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

Step 5.6.3.10.3

Add $2$ and $1$ .

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

Step 5.6.3.11

Multiply $- 1$ by $3$ .

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

Step 5.6.3.12

Rewrite using the commutative property of multiplication.

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

Step 5.6.3.13

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

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Step 5.6.3.13.1

Move $λ$ .

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

Step 5.6.3.13.2

Multiply $λ$ by $λ$ .

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

Step 5.6.3.14

Multiply $- 1$ by $- 3$ .

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

Step 5.6.3.15

Multiply $- 1$ by $1$ .

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

Step 5.6.4

Subtract $3 λ^{3}$ from $- λ^{3}$ .

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

Step 5.6.5

Add $3 λ^{2}$ and $3 λ^{2}$ .

$p (λ) = - 4 λ^{3} + 6 λ^{2} - 3 λ + 1 + λ^{4} - λ$

Step 5.6.6

Subtract $λ$ from $- 3 λ$ .

$p (λ) = - 4 λ^{3} + 6 λ^{2} - 4 λ + 1 + λ^{4}$

Step 5.6.7

Move $1$ .

$p (λ) = - 4 λ^{3} + 6 λ^{2} - 4 λ + λ^{4} + 1$

Step 5.6.8

Move $- 4 λ$ .

$p (λ) = - 4 λ^{3} + 6 λ^{2} + λ^{4} - 4 λ + 1$

Step 5.6.9

Move $6 λ^{2}$ .

$p (λ) = - 4 λ^{3} + λ^{4} + 6 λ^{2} - 4 λ + 1$

Step 5.6.10

Reorder $- 4 λ^{3}$ and $λ^{4}$ .

$p (λ) = λ^{4} - 4 λ^{3} + 6 λ^{2} - 4 λ + 1$

Step 6

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

$λ^{4} - 4 λ^{3} + 6 λ^{2} - 4 λ + 1 = 0$

Step 7

Solve for $λ$ .

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Step 7.1

Factor $λ^{4} - 4 λ^{3} + 6 λ^{2} - 4 λ + 1$ using the binomial theorem.

${(1 - λ)}^{4} = 0$

Step 7.2

Set the $1 - λ$ equal to $0$ .

$1 - λ = 0$

Step 7.3

Solve for $λ$ .

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Step 7.3.1

Subtract $1$ from both sides of the equation.

$- λ = - 1$

Step 7.3.2

Divide each term in $- λ = - 1$ by $- 1$ and simplify.

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Step 7.3.2.1

Divide each term in $- λ = - 1$ by $- 1$ .

$\frac{- λ}{- 1} = \frac{- 1}{- 1}$

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{λ}{1} = \frac{- 1}{- 1}$

Step 7.3.2.2.2

Divide $λ$ by $1$ .

$λ = \frac{- 1}{- 1}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

Divide $- 1$ by $- 1$ .

$λ = 1$