Linear Algebra Examples

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

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 $[\begin{array}{c} 3 & 1 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$ for $A$ .

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

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 $I_{4}$ .

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

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.

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

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

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

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.5

Multiply $- λ \cdot 0$ .

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Step 4.1.2.5.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.5.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.6

Multiply $- 1$ by $1$ .

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

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.9

Multiply $- λ \cdot 0$ .

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Step 4.1.2.9.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.9.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.10

Multiply $- λ \cdot 0$ .

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Step 4.1.2.10.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.10.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.11

Multiply $- 1$ by $1$ .

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

Step 4.1.2.12

Multiply $- λ \cdot 0$ .

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Step 4.1.2.12.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.12.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.13

Multiply $- λ \cdot 0$ .

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Step 4.1.2.13.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.13.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.14

Multiply $- λ \cdot 0$ .

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Step 4.1.2.14.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.14.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.15

Multiply $- λ \cdot 0$ .

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Step 4.1.2.15.1

Multiply $0$ by $- 1$ .

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

Step 4.1.2.15.2

Multiply $0$ by $λ$ .

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

Step 4.1.2.16

Multiply $- 1$ by $1$ .

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

Step 4.2

Add the corresponding elements.

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

Step 4.3

Simplify each element.

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Step 4.3.1

Add $1$ and $0$ .

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

Step 4.3.2

Add $0$ and $0$ .

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

Step 4.3.3

Add $0$ and $0$ .

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

Step 4.3.4

Add $1$ and $0$ .

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

Step 4.3.5

Add $0$ and $0$ .

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

Step 4.3.6

Add $0$ and $0$ .

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

Step 4.3.7

Add $0$ and $0$ .

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

Step 4.3.8

Add $0$ and $0$ .

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

Step 4.3.9

Subtract $λ$ from $0$ .

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

Step 4.3.10

Add $0$ and $0$ .

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

Step 4.3.11

Add $0$ and $0$ .

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

Step 4.3.12

Add $0$ and $0$ .

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

Step 4.3.13

Add $0$ and $0$ .

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

Step 4.3.14

Subtract $λ$ from $0$ .

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

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

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

Step 5.1.8

Multiply element $a_{33}$ by its cofactor.

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

Step 5.1.9

The minor for $a_{34}$ is the determinant with row $3$ and column $4$ deleted.

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

Step 5.1.10

Multiply element $a_{34}$ by its cofactor.

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

Step 5.1.11

Add the terms together.

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

Step 5.2

Multiply $0$ by $| \begin{array}{c} 1 & 0 & 0 \\ 3 - λ & 0 & 0 \\ 0 & 0 & - λ \end{array} |$ .

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

Step 5.3

Multiply $0$ by $| \begin{array}{c} 3 - λ & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & - λ \end{array} |$ .

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

Step 5.4

Multiply $0$ by $| \begin{array}{c} 3 - λ & 1 & 0 \\ 1 & 3 - λ & 0 \\ 0 & 0 & 0 \end{array} |$ .

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

Step 5.5

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

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

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

Step 5.5.1.4

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

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

Step 5.5.1.5

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

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

Step 5.5.1.6

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

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

Step 5.5.1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

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

Step 5.5.1.8

Multiply element $a_{33}$ by its cofactor.

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

Step 5.5.1.9

Add the terms together.

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

Step 5.5.2

Multiply $0$ by $| \begin{array}{c} 1 & 0 \\ 3 - λ & 0 \end{array} |$ .

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

Step 5.5.3

Multiply $0$ by $| \begin{array}{c} 3 - λ & 0 \\ 1 & 0 \end{array} |$ .

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

Step 5.5.4

Evaluate $| \begin{array}{c} 3 - λ & 1 \\ 1 & 3 - λ \end{array} |$ .

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Step 5.5.4.1

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

$p (λ) = 0 + 0 - λ (0 + 0 - λ ((3 - λ) (3 - λ) - 1 \cdot 1)) + 0$

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 $(3 - λ) (3 - λ)$ using the FOIL Method.

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Step 5.5.4.2.1.1.1

Apply the distributive property.

$p (λ) = 0 + 0 - λ (0 + 0 - λ (3 (3 - λ) - λ (3 - λ) - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.1.2

Apply the distributive property.

$p (λ) = 0 + 0 - λ (0 + 0 - λ (3 \cdot 3 + 3 (- λ) - λ (3 - λ) - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.1.3

Apply the distributive property.

$p (λ) = 0 + 0 - λ (0 + 0 - λ (3 \cdot 3 + 3 (- λ) - λ \cdot 3 - λ (- λ) - 1 \cdot 1)) + 0$

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 $3$ by $3$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 + 3 (- λ) - λ \cdot 3 - λ (- λ) - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.1.2

Multiply $- 1$ by $3$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - λ \cdot 3 - λ (- λ) - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.1.3

Multiply $3$ by $- 1$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - 3 λ - λ (- λ) - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - 3 λ - 1 \cdot - 1 λ \cdot λ - 1 \cdot 1)) + 0$

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 $λ$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - 3 λ - 1 \cdot - 1 (λ \cdot λ) - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - 3 λ - 1 \cdot - 1 λ^{2} - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - 3 λ + 1 λ^{2} - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 3 λ - 3 λ + λ^{2} - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.2.2

Subtract $3 λ$ from $- 3 λ$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 6 λ + λ^{2} - 1 \cdot 1)) + 0$

Step 5.5.4.2.1.3

Multiply $- 1$ by $1$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (9 - 6 λ + λ^{2} - 1)) + 0$

Step 5.5.4.2.2

Subtract $1$ from $9$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (- 6 λ + λ^{2} + 8)) + 0$

Step 5.5.4.2.3

Reorder $- 6 λ$ and $λ^{2}$ .

$p (λ) = 0 + 0 - λ (0 + 0 - λ (λ^{2} - 6 λ + 8)) + 0$

Step 5.5.5

Simplify the determinant.

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Step 5.5.5.1

Combine the opposite terms in $0 + 0 - λ (λ^{2} - 6 λ + 8)$ .

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Step 5.5.5.1.1

Add $0$ and $0$ .

$p (λ) = 0 + 0 - λ (0 - λ (λ^{2} - 6 λ + 8)) + 0$

Step 5.5.5.1.2

Subtract $λ (λ^{2} - 6 λ + 8)$ from $0$ .

$p (λ) = 0 + 0 - λ (- λ (λ^{2} - 6 λ + 8)) + 0$

Step 5.5.5.2

Apply the distributive property.

$p (λ) = 0 + 0 - λ (- λ \cdot λ^{2} - λ (- 6 λ) - λ \cdot 8) + 0$

Step 5.5.5.3

Simplify.

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Step 5.5.5.3.1

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

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Step 5.5.5.3.1.1

Move $λ^{2}$ .

$p (λ) = 0 + 0 - λ (- (λ^{2} λ) - λ (- 6 λ) - λ \cdot 8) + 0$

Step 5.5.5.3.1.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.5.5.3.1.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = 0 + 0 - λ (- (λ^{2} λ^{1}) - λ (- 6 λ) - λ \cdot 8) + 0$

Step 5.5.5.3.1.2.2

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

$p (λ) = 0 + 0 - λ (- λ^{2 + 1} - λ (- 6 λ) - λ \cdot 8) + 0$

Step 5.5.5.3.1.3

Add $2$ and $1$ .

$p (λ) = 0 + 0 - λ (- λ^{3} - λ (- 6 λ) - λ \cdot 8) + 0$

Step 5.5.5.3.2

Rewrite using the commutative property of multiplication.

$p (λ) = 0 + 0 - λ (- λ^{3} - 1 \cdot - 6 λ \cdot λ - λ \cdot 8) + 0$

Step 5.5.5.3.3

Multiply $8$ by $- 1$ .

$p (λ) = 0 + 0 - λ (- λ^{3} - 1 \cdot - 6 λ \cdot λ - 8 λ) + 0$

Step 5.5.5.4

Simplify each term.

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Step 5.5.5.4.1

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

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Step 5.5.5.4.1.1

Move $λ$ .

$p (λ) = 0 + 0 - λ (- λ^{3} - 1 \cdot - 6 (λ \cdot λ) - 8 λ) + 0$

Step 5.5.5.4.1.2

Multiply $λ$ by $λ$ .

$p (λ) = 0 + 0 - λ (- λ^{3} - 1 \cdot - 6 λ^{2} - 8 λ) + 0$

Step 5.5.5.4.2

Multiply $- 1$ by $- 6$ .

$p (λ) = 0 + 0 - λ (- λ^{3} + 6 λ^{2} - 8 λ) + 0$

Step 5.6

Simplify the determinant.

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Step 5.6.1

Combine the opposite terms in $0 + 0 - λ (- λ^{3} + 6 λ^{2} - 8 λ) + 0$ .

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Step 5.6.1.1

Add $0$ and $0$ .

$p (λ) = 0 - λ (- λ^{3} + 6 λ^{2} - 8 λ) + 0$

Step 5.6.1.2

Subtract $λ (- λ^{3} + 6 λ^{2} - 8 λ)$ from $0$ .

$p (λ) = - λ (- λ^{3} + 6 λ^{2} - 8 λ) + 0$

Step 5.6.1.3

Add $- λ (- λ^{3} + 6 λ^{2} - 8 λ)$ and $0$ .

$p (λ) = - λ (- λ^{3} + 6 λ^{2} - 8 λ)$

Step 5.6.2

Apply the distributive property.

$p (λ) = - λ (- λ^{3}) - λ (6 λ^{2}) - λ (- 8 λ)$

Step 5.6.3

Simplify.

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Step 5.6.3.1

Rewrite using the commutative property of multiplication.

$p (λ) = - 1 \cdot - 1 λ \cdot λ^{3} - λ (6 λ^{2}) - λ (- 8 λ)$

Step 5.6.3.2

Rewrite using the commutative property of multiplication.

$p (λ) = - 1 \cdot - 1 λ \cdot λ^{3} - 1 \cdot 6 λ \cdot λ^{2} - λ (- 8 λ)$

Step 5.6.3.3

Rewrite using the commutative property of multiplication.

$p (λ) = - 1 \cdot - 1 λ \cdot λ^{3} - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4

Simplify each term.

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Step 5.6.4.1

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

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Step 5.6.4.1.1

Move $λ^{3}$ .

$p (λ) = - 1 \cdot - 1 (λ^{3} λ) - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.1.2

Multiply $λ^{3}$ by $λ$ .

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Step 5.6.4.1.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 1 \cdot - 1 (λ^{3} λ^{1}) - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.1.2.2

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

$p (λ) = - 1 \cdot - 1 λ^{3 + 1} - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.1.3

Add $3$ and $1$ .

$p (λ) = - 1 \cdot - 1 λ^{4} - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.2

Multiply $- 1$ by $- 1$ .

$p (λ) = 1 λ^{4} - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.3

Multiply $λ^{4}$ by $1$ .

$p (λ) = λ^{4} - 1 \cdot 6 λ \cdot λ^{2} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.4

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

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Step 5.6.4.4.1

Move $λ^{2}$ .

$p (λ) = λ^{4} - 1 \cdot 6 (λ^{2} λ) - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.4.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.6.4.4.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = λ^{4} - 1 \cdot 6 (λ^{2} λ^{1}) - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.4.2.2

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

$p (λ) = λ^{4} - 1 \cdot 6 λ^{2 + 1} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.4.3

Add $2$ and $1$ .

$p (λ) = λ^{4} - 1 \cdot 6 λ^{3} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.5

Multiply $- 1$ by $6$ .

$p (λ) = λ^{4} - 6 λ^{3} - 1 \cdot - 8 λ \cdot λ$

Step 5.6.4.6

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

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Step 5.6.4.6.1

Move $λ$ .

$p (λ) = λ^{4} - 6 λ^{3} - 1 \cdot - 8 (λ \cdot λ)$

Step 5.6.4.6.2

Multiply $λ$ by $λ$ .

$p (λ) = λ^{4} - 6 λ^{3} - 1 \cdot - 8 λ^{2}$

Step 5.6.4.7

Multiply $- 1$ by $- 8$ .

$p (λ) = λ^{4} - 6 λ^{3} + 8 λ^{2}$

Step 6

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

$λ^{4} - 6 λ^{3} + 8 λ^{2} = 0$

Step 7

Solve for $λ$ .

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Step 7.1

Factor the left side of the equation.

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Step 7.1.1

Factor $λ^{2}$ out of $λ^{4} - 6 λ^{3} + 8 λ^{2}$ .

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Step 7.1.1.1

Factor $λ^{2}$ out of $λ^{4}$ .

$λ^{2} λ^{2} - 6 λ^{3} + 8 λ^{2} = 0$

Step 7.1.1.2

Factor $λ^{2}$ out of $- 6 λ^{3}$ .

$λ^{2} λ^{2} + λ^{2} (- 6 λ) + 8 λ^{2} = 0$

Step 7.1.1.3

Factor $λ^{2}$ out of $8 λ^{2}$ .

$λ^{2} λ^{2} + λ^{2} (- 6 λ) + λ^{2} \cdot 8 = 0$

Step 7.1.1.4

Factor $λ^{2}$ out of $λ^{2} λ^{2} + λ^{2} (- 6 λ)$ .

$λ^{2} (λ^{2} - 6 λ) + λ^{2} \cdot 8 = 0$

Step 7.1.1.5

Factor $λ^{2}$ out of $λ^{2} (λ^{2} - 6 λ) + λ^{2} \cdot 8$ .

$λ^{2} (λ^{2} - 6 λ + 8) = 0$

Step 7.1.2

Factor.

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Step 7.1.2.1

Factor $λ^{2} - 6 λ + 8$ using the AC method.

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Step 7.1.2.1.1

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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 7.1.2.1.2

Write the factored form using these integers.

$λ^{2} ((λ - 4) (λ - 2)) = 0$

Step 7.1.2.2

Remove unnecessary parentheses.

$λ^{2} (λ - 4) (λ - 2) = 0$

Step 7.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$λ^{2} = 0$

$λ - 4 = 0$

$λ - 2 = 0$

Step 7.3

Set $λ^{2}$ equal to $0$ and solve for $λ$ .

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Step 7.3.1

Set $λ^{2}$ equal to $0$ .

$λ^{2} = 0$

Step 7.3.2

Solve $λ^{2} = 0$ for $λ$ .

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Step 7.3.2.1

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

$λ = \pm \sqrt{0}$

Step 7.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$λ = \pm \sqrt{0^{2}}$

Step 7.3.2.2.2

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

$λ = \pm 0$

Step 7.3.2.2.3

Plus or minus $0$ is $0$ .

$λ = 0$

Step 7.4

Set $λ - 4$ equal to $0$ and solve for $λ$ .

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Step 7.4.1

Set $λ - 4$ equal to $0$ .

$λ - 4 = 0$

Step 7.4.2

Add $4$ to both sides of the equation.

$λ = 4$

Step 7.5

Set $λ - 2$ equal to $0$ and solve for $λ$ .

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Step 7.5.1

Set $λ - 2$ equal to $0$ .

$λ - 2 = 0$

Step 7.5.2

Add $2$ to both sides of the equation.

$λ = 2$

Step 7.6

The final solution is all the values that make $λ^{2} (λ - 4) (λ - 2) = 0$ true.

$λ = 0, 4, 2$