Linear Algebra Examples

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

Step 1

Find the eigenvalues.

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Step 1.1

Set up the formula to find the characteristic equation

$p (λ)$ .

$p (λ) = determinant (A - λ I_{4})$

Step 1.2

The identity matrix or unit matrix of size

$4$ is the

$4 \times 4$ square matrix with ones on the main diagonal and zeros elsewhere.

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

Step 1.3

Substitute the known values into

$p (λ) = determinant (A - λ I_{4})$ .

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Step 1.3.1

Substitute

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

$A$ .

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

Step 1.3.2

Substitute

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

$I_{4}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply

$- λ$ by each element of the matrix.

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

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply

$- 1$ by

$1$ .

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

Step 1.4.1.2.2

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.2.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.3

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.3.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.4

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.4.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.4.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.5

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.5.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.5.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.6

Multiply

$- 1$ by

$1$ .

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

Step 1.4.1.2.7

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.7.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.7.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.8

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.8.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.8.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.9

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.9.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.9.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.10

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.10.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.10.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.11

Multiply

$- 1$ by

$1$ .

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

Step 1.4.1.2.12

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.12.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.12.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.13

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.13.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.13.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.14

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.14.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.14.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.15

Multiply

$- λ \cdot 0$ .

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Step 1.4.1.2.15.1

Multiply

$0$ by

$- 1$ .

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

Step 1.4.1.2.15.2

Multiply

$0$ by

$λ$ .

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

Step 1.4.1.2.16

Multiply

$- 1$ by

$1$ .

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

Step 1.4.2

Add the corresponding elements.

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

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Subtract

$λ$ from

$0$ .

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

Step 1.4.3.2

Add

$1$ and

$0$ .

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

Step 1.4.3.3

Add

$0$ and

$0$ .

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

Step 1.4.3.4

Add

$- 1$ and

$0$ .

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

Step 1.4.3.5

Add

$1$ and

$0$ .

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

Step 1.4.3.6

Subtract

$λ$ from

$0$ .

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

Step 1.4.3.7

Add

$- 1$ and

$0$ .

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

Step 1.4.3.8

Add

$0$ and

$0$ .

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

Step 1.4.3.9

Add

$0$ and

$0$ .

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

Step 1.4.3.10

Add

$- 1$ and

$0$ .

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

Step 1.4.3.11

Subtract

$λ$ from

$0$ .

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

Step 1.4.3.12

Add

$- 1$ and

$0$ .

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

Step 1.4.3.13

Add

$- 1$ and

$0$ .

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

Step 1.4.3.14

Add

$0$ and

$0$ .

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

Step 1.4.3.15

Add

$- 1$ and

$0$ .

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

Step 1.4.3.16

Subtract

$λ$ from

$0$ .

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

Step 1.5

Find the determinant.

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Step 1.5.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$1$ by its cofactor and add.

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Step 1.5.1.1

Consider the corresponding sign chart.

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

Step 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 1.5.1.3

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

Step 1.5.1.4

Multiply element

$a_{11}$ by its cofactor.

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

Step 1.5.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

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

Step 1.5.1.6

Multiply element

$a_{12}$ by its cofactor.

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

Step 1.5.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

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

Step 1.5.1.8

Multiply element

$a_{13}$ by its cofactor.

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

Step 1.5.1.9

The minor for

$a_{14}$ is the determinant with row

$1$ and column

$4$ deleted.

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

Step 1.5.1.10

Multiply element

$a_{14}$ by its cofactor.

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

Step 1.5.1.11

Add the terms together.

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

Step 1.5.2

Multiply

$0$ by

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

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

Step 1.5.3

Evaluate

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

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Step 1.5.3.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$1$ by its cofactor and add.

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Step 1.5.3.1.1

Consider the corresponding sign chart.

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

Step 1.5.3.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 1.5.3.1.3

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

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

Step 1.5.3.1.4

Multiply element

$a_{11}$ by its cofactor.

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

Step 1.5.3.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

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

Step 1.5.3.1.6

Multiply element

$a_{12}$ by its cofactor.

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

Step 1.5.3.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

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

Step 1.5.3.1.8

Multiply element

$a_{13}$ by its cofactor.

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

Step 1.5.3.1.9

Add the terms together.

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

Step 1.5.3.2

Multiply

$0$ by

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

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

Step 1.5.3.3

Evaluate

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

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Step 1.5.3.3.1

The determinant of a

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

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

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

Step 1.5.3.3.2

Simplify each term.

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Step 1.5.3.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.3.3.2.2

Multiply

$λ$ by

$λ$ by adding the exponents.

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Step 1.5.3.3.2.2.1

Move

$λ$ .

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

Step 1.5.3.3.2.2.2

Multiply

$λ$ by

$λ$ .

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

Step 1.5.3.3.2.3

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.3.3.2.4

Multiply

$λ^{2}$ by

$1$ .

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

Step 1.5.3.3.2.5

Multiply

$- - - 1$ .

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Step 1.5.3.3.2.5.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.3.3.2.5.2

Multiply

$- 1$ by

$1$ .

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

Step 1.5.3.4

Evaluate

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

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Step 1.5.3.4.1

The determinant of a

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

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

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

Step 1.5.3.4.2

Simplify the determinant.

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Step 1.5.3.4.2.1

Simplify each term.

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Step 1.5.3.4.2.1.1

Multiply

$- - λ$ .

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Step 1.5.3.4.2.1.1.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.3.4.2.1.1.2

Multiply

$λ$ by

$1$ .

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

Step 1.5.3.4.2.1.2

Multiply

$0$ by

$- 1$ .

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

Step 1.5.3.4.2.2

Add

$λ$ and

$0$ .

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

Step 1.5.3.5

Simplify the determinant.

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Step 1.5.3.5.1

Add

$- λ (λ^{2} - 1) + 1 λ$ and

$0$ .

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

Step 1.5.3.5.2

Simplify each term.

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Step 1.5.3.5.2.1

Apply the distributive property.

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

Step 1.5.3.5.2.2

Multiply

$λ$ by

$λ^{2}$ by adding the exponents.

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Step 1.5.3.5.2.2.1

Move

$λ^{2}$ .

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

Step 1.5.3.5.2.2.2

Multiply

$λ^{2}$ by

$λ$ .

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Step 1.5.3.5.2.2.2.1

Raise

$λ$ to the power of

$1$ .

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

Step 1.5.3.5.2.2.2.2

Use the power rule

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

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

Step 1.5.3.5.2.2.3

Add

$2$ and

$1$ .

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

Step 1.5.3.5.2.3

Multiply

$- λ \cdot - 1$ .

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Step 1.5.3.5.2.3.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.3.5.2.3.2

Multiply

$λ$ by

$1$ .

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

Step 1.5.3.5.2.4

Multiply

$λ$ by

$1$ .

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

Step 1.5.3.5.3

Add

$λ$ and

$λ$ .

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

Step 1.5.4

Evaluate

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

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Step 1.5.4.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$2$ by its cofactor and add.

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Step 1.5.4.1.1

Consider the corresponding sign chart.

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

Step 1.5.4.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 1.5.4.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

Step 1.5.4.1.4

Multiply element

$a_{21}$ by its cofactor.

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

Step 1.5.4.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

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

Step 1.5.4.1.6

Multiply element

$a_{22}$ by its cofactor.

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

Step 1.5.4.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

Step 1.5.4.1.8

Multiply element

$a_{23}$ by its cofactor.

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

Step 1.5.4.1.9

Add the terms together.

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

Step 1.5.4.2

Multiply

$0$ by

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

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

Step 1.5.4.3

Evaluate

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

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Step 1.5.4.3.1

The determinant of a

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

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

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

Step 1.5.4.3.2

Simplify the determinant.

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Step 1.5.4.3.2.1

Simplify each term.

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Step 1.5.4.3.2.1.1

Multiply

$- λ$ by

$1$ .

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

Step 1.5.4.3.2.1.2

Multiply

$- - 0$ .

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Step 1.5.4.3.2.1.2.1

Multiply

$- 1$ by

$0$ .

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

Step 1.5.4.3.2.1.2.2

Multiply

$- 1$ by

$0$ .

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

Step 1.5.4.3.2.2

Add

$- λ$ and

$0$ .

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

Step 1.5.4.4

Evaluate

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

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Step 1.5.4.4.1

The determinant of a

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

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

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

Step 1.5.4.4.2

Simplify the determinant.

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Step 1.5.4.4.2.1

Simplify each term.

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Step 1.5.4.4.2.1.1

Multiply

$- 1$ by

$1$ .

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

Step 1.5.4.4.2.1.2

Multiply

$- - - 1$ .

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Step 1.5.4.4.2.1.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.4.4.2.1.2.2

Multiply

$- 1$ by

$1$ .

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

Step 1.5.4.4.2.2

Subtract

$1$ from

$- 1$ .

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

Step 1.5.4.5

Simplify the determinant.

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Step 1.5.4.5.1

Subtract

$λ (- λ)$ from

$0$ .

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

Step 1.5.4.5.2

Simplify each term.

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Step 1.5.4.5.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.4.5.2.2

Multiply

$λ$ by

$λ$ by adding the exponents.

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Step 1.5.4.5.2.2.1

Move

$λ$ .

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

Step 1.5.4.5.2.2.2

Multiply

$λ$ by

$λ$ .

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

Step 1.5.4.5.2.3

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.4.5.2.4

Multiply

$λ^{2}$ by

$1$ .

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

Step 1.5.4.5.2.5

Multiply

$- 2$ by

$1$ .

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

Step 1.5.5

Evaluate

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

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Step 1.5.5.1

Choose the row or column with the most

$0$ elements. If there are no

$0$ elements choose any row or column. Multiply every element in row

$2$ by its cofactor and add.

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Step 1.5.5.1.1

Consider the corresponding sign chart.

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

Step 1.5.5.1.2

The cofactor is the minor with the sign changed if the indices match a

$-$ position on the sign chart.

Step 1.5.5.1.3

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

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

Step 1.5.5.1.4

Multiply element

$a_{21}$ by its cofactor.

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

Step 1.5.5.1.5

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

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

Step 1.5.5.1.6

Multiply element

$a_{22}$ by its cofactor.

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

Step 1.5.5.1.7

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

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

Step 1.5.5.1.8

Multiply element

$a_{23}$ by its cofactor.

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

Step 1.5.5.1.9

Add the terms together.

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

Step 1.5.5.2

Multiply

$0$ by

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

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

Step 1.5.5.3

Evaluate

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

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Step 1.5.5.3.1

The determinant of a

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

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

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

Step 1.5.5.3.2

Simplify the determinant.

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Step 1.5.5.3.2.1

Simplify each term.

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Step 1.5.5.3.2.1.1

Multiply

$- 1$ by

$1$ .

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

Step 1.5.5.3.2.1.2

Multiply

$- - - 1$ .

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Step 1.5.5.3.2.1.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.5.3.2.1.2.2

Multiply

$- 1$ by

$1$ .

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

Step 1.5.5.3.2.2

Subtract

$1$ from

$- 1$ .

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

Step 1.5.5.4

Evaluate

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

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Step 1.5.5.4.1

The determinant of a

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

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

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

Step 1.5.5.4.2

Simplify the determinant.

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Step 1.5.5.4.2.1

Simplify each term.

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Step 1.5.5.4.2.1.1

Multiply

$0$ by

$1$ .

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

Step 1.5.5.4.2.1.2

Multiply

$- - λ$ .

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Step 1.5.5.4.2.1.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.5.4.2.1.2.2

Multiply

$λ$ by

$1$ .

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

Step 1.5.5.4.2.2

Subtract

$λ$ from

$0$ .

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

Step 1.5.5.5

Simplify the determinant.

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Step 1.5.5.5.1

Subtract

$1 \cdot - 2$ from

$0$ .

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

Step 1.5.5.5.2

Simplify each term.

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Step 1.5.5.5.2.1

Multiply

$- 1$ by

$- 2$ .

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

Step 1.5.5.5.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.5.5.2.3

Multiply

$λ$ by

$λ$ by adding the exponents.

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Step 1.5.5.5.2.3.1

Move

$λ$ .

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

Step 1.5.5.5.2.3.2

Multiply

$λ$ by

$λ$ .

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

Step 1.5.5.5.3

Reorder

$2$ and

$- λ^{2}$ .

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

Step 1.5.6

Simplify the determinant.

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Step 1.5.6.1

Add

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

$0$ .

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

Step 1.5.6.2

Simplify each term.

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Step 1.5.6.2.1

Apply the distributive property.

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

Step 1.5.6.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.6.2.3

Rewrite using the commutative property of multiplication.

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

Step 1.5.6.2.4

Simplify each term.

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Step 1.5.6.2.4.1

Multiply

$λ$ by

$λ^{3}$ by adding the exponents.

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Step 1.5.6.2.4.1.1

Move

$λ^{3}$ .

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

Step 1.5.6.2.4.1.2

Multiply

$λ^{3}$ by

$λ$ .

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Step 1.5.6.2.4.1.2.1

Raise

$λ$ to the power of

$1$ .

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

Step 1.5.6.2.4.1.2.2

Use the power rule

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

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

Step 1.5.6.2.4.1.3

Add

$3$ and

$1$ .

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

Step 1.5.6.2.4.2

Multiply

$- 1$ by

$- 1$ .

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

Step 1.5.6.2.4.3

Multiply

$λ^{4}$ by

$1$ .

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

Step 1.5.6.2.4.4

Multiply

$λ$ by

$λ$ by adding the exponents.

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Step 1.5.6.2.4.4.1

Move

$λ$ .

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

Step 1.5.6.2.4.4.2

Multiply

$λ$ by

$λ$ .

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

Step 1.5.6.2.4.5

Multiply

$- 1$ by

$2$ .

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

Step 1.5.6.2.5

Apply the distributive property.

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

Step 1.5.6.2.6

Rewrite

$- 1 λ^{2}$ as

$- λ^{2}$ .

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

Step 1.5.6.2.7

Multiply

$- 1$ by

$- 2$ .

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

Step 1.5.6.2.8

Multiply

$- λ^{2} + 2$ by

$1$ .

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

Step 1.5.6.3

Subtract

$λ^{2}$ from

$- 2 λ^{2}$ .

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

Step 1.5.6.4

Subtract

$λ^{2}$ from

$- 3 λ^{2}$ .

$p (λ) = λ^{4} - 4 λ^{2} + 2 + 2$

Step 1.5.6.5

Add

$2$ and

$2$ .

$p (λ) = λ^{4} - 4 λ^{2} + 4$

Step 1.6

Set the characteristic polynomial equal to

$0$ to find the eigenvalues

$λ$ .

$λ^{4} - 4 λ^{2} + 4 = 0$

Step 1.7

Solve for

$λ$ .

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Step 1.7.1

Substitute

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

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

$u = λ^{2}$

Step 1.7.2

Factor using the perfect square rule.

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Step 1.7.2.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 1.7.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 1.7.2.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 2 + 2^{2} = 0$

Step 1.7.2.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = u$ and

$b = 2$ .

${(u - 2)}^{2} = 0$

Step 1.7.3

Set the

$u - 2$ equal to

$0$ .

$u - 2 = 0$

Step 1.7.4

Add

$2$ to both sides of the equation.

$u = 2$

Step 1.7.5

Substitute the real value of

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

$λ^{2} = 2$

Step 1.7.6

Solve the equation for

$λ$ .

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Step 1.7.6.1

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

$λ = \pm \sqrt{2}$

Step 1.7.6.2

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

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Step 1.7.6.2.1

First, use the positive value of the

$\pm$ to find the first solution.

$λ = \sqrt{2}$

Step 1.7.6.2.2

Next, use the negative value of the

$\pm$ to find the second solution.

$λ = - \sqrt{2}$

Step 1.7.6.2.3

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

$λ = \sqrt{2}, - \sqrt{2}$

Step 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where

$N$ is the null space and

$I$ is the identity matrix.

$ε_{A} = N (A - λ I_{4})$

Step 3

Find the eigenvector using the eigenvalue

$λ = \sqrt{2}$ .

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Step 3.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] - \sqrt{2} [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply

$- \sqrt{2}$ by each element of the matrix.

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply

$- 1$ by

$1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.2.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 \sqrt{2} & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.2.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.3.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 \sqrt{2} & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.3.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.4.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \sqrt{2} \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.4.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.5

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.5.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 \sqrt{2} & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.5.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.6

Multiply

$- 1$ by

$1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.7

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.7.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 \sqrt{2} & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.7.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.8

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.8.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \sqrt{2} \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.8.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.9

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.9.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 \sqrt{2} & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.9.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.10

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.10.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 \sqrt{2} & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.10.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} \cdot 1 & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.11

Multiply

$- 1$ by

$1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & - \sqrt{2} \cdot 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.12

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.12.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \sqrt{2} \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.12.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.13

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.13.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ 0 \sqrt{2} & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.13.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.14

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.14.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ 0 & 0 \sqrt{2} & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.14.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ 0 & 0 & - \sqrt{2} \cdot 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.15

Multiply

$- \sqrt{2} \cdot 0$ .

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Step 3.2.1.2.15.1

Multiply

$0$ by

$- 1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ 0 & 0 & 0 \sqrt{2} & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.15.2

Multiply

$0$ by

$\sqrt{2}$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} - \sqrt{2} & 0 & 0 & 0 \\ 0 & - \sqrt{2} & 0 & 0 \\ 0 & 0 & - \sqrt{2} & 0 \\ 0 & 0 & 0 & - \sqrt{2} \cdot 1 \end{array}]$

Step 3.2.1.2.16

Multiply

$- 1$ by

$1$ .

Step 3.2.2

Add the corresponding elements.

$[\begin{array}{c} 0 - \sqrt{2} & 1 + 0 & 0 + 0 & - 1 + 0 \\ 1 + 0 & 0 - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3

Simplify each element.

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Step 3.2.3.1

Subtract

$\sqrt{2}$ from

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 + 0 & 0 + 0 & - 1 + 0 \\ 1 + 0 & 0 - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.2

Add

$1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 + 0 & - 1 + 0 \\ 1 + 0 & 0 - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.3

Add

$0$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 + 0 \\ 1 + 0 & 0 - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.4

Add

$- 1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 + 0 & 0 - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.5

Add

$1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & 0 - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.6

Subtract

$\sqrt{2}$ from

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.7

Add

$- 1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.8

Add

$0$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 + 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.9

Add

$0$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 + 0 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.10

Add

$- 1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & 0 - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.11

Subtract

$\sqrt{2}$ from

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.12

Add

$- 1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.13

Add

$- 1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 \\ - 1 & 0 + 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.14

Add

$0$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 \\ - 1 & 0 & - 1 + 0 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.15

Add

$- 1$ and

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 \\ - 1 & 0 & - 1 & 0 - \sqrt{2} \end{array}]$

Step 3.2.3.16

Subtract

$\sqrt{2}$ from

$0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 \\ 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 \\ - 1 & 0 & - 1 & - \sqrt{2} \end{array}]$

Step 3.3

Find the null space when

$λ = \sqrt{2}$ .

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Step 3.3.1

Write as an augmented matrix for

$A x = 0$ .

$[\begin{array}{c} - \sqrt{2} & 1 & 0 & - 1 & 0 \\ 1 & - \sqrt{2} & - 1 & 0 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & - \sqrt{2} & 0 \end{array}]$

Step 3.3.2

Find the reduced row echelon form.

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Step 3.3.2.1

Multiply each element of

$R_{1}$ by

$- \frac{1}{\sqrt{2}}$ to make the entry at

$1, 1$ a

$1$ .

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Step 3.3.2.1.1

Multiply each element of

$R_{1}$ by

$- \frac{1}{\sqrt{2}}$ to make the entry at

$1, 1$ a

$1$ .

$[\begin{array}{c} - \frac{1}{\sqrt{2}} (- \sqrt{2}) & - \frac{1}{\sqrt{2}} \cdot 1 & - \frac{1}{\sqrt{2}} \cdot 0 & - \frac{1}{\sqrt{2}} \cdot - 1 & - \frac{1}{\sqrt{2}} \cdot 0 \\ 1 & - \sqrt{2} & - 1 & 0 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & - \sqrt{2} & 0 \end{array}]$

Step 3.3.2.1.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 1 & - \sqrt{2} & - 1 & 0 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & - \sqrt{2} & 0 \end{array}]$

Step 3.3.2.2

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Step 3.3.2.2.1

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 1 - 1 & - \sqrt{2} + \frac{\sqrt{2}}{2} & - 1 - 0 & 0 - \frac{\sqrt{2}}{2} & 0 - 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & - \sqrt{2} & 0 \end{array}]$

Step 3.3.2.2.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & - \sqrt{2} & 0 \end{array}]$

Step 3.3.2.3

Perform the row operation

$R_{4} = R_{4} + R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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Step 3.3.2.3.1

Perform the row operation

$R_{4} = R_{4} + R_{1}$ to make the entry at

$4, 1$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ - 1 + 1 \cdot 1 & 0 - \frac{\sqrt{2}}{2} & - 1 + 0 & - \sqrt{2} + \frac{\sqrt{2}}{2} & 0 + 0 \end{array}]$

Step 3.3.2.3.2

Simplify

$R_{4}$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 3.3.2.4

Multiply each element of

$R_{2}$ by

$- \frac{2}{\sqrt{2}}$ to make the entry at

$2, 2$ a

$1$ .

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Step 3.3.2.4.1

Multiply each element of

$R_{2}$ by

$- \frac{2}{\sqrt{2}}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ - \frac{2}{\sqrt{2}} \cdot 0 & - \frac{2}{\sqrt{2}} (- \frac{\sqrt{2}}{2}) & - \frac{2}{\sqrt{2}} \cdot - 1 & - \frac{2}{\sqrt{2}} (- \frac{\sqrt{2}}{2}) & - \frac{2}{\sqrt{2}} \cdot 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 3.3.2.4.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 & - 1 & - \sqrt{2} & - 1 & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 3.3.2.5

Perform the row operation

$R_{3} = R_{3} + R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Step 3.3.2.5.1

Perform the row operation

$R_{3} = R_{3} + R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 + 0 & - 1 + 1 \cdot 1 & - \sqrt{2} + \sqrt{2} & - 1 + 1 \cdot 1 & 0 + 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 3.3.2.5.2

Simplify

$R_{3}$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{\sqrt{2}}{2} & - 1 & - \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 3.3.2.6

Perform the row operation

$R_{4} = R_{4} + \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Step 3.3.2.6.1

Perform the row operation

$R_{4} = R_{4} + \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 + \frac{\sqrt{2}}{2} \cdot 0 & - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 1 & - 1 + \frac{\sqrt{2}}{2} \sqrt{2} & - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 1 & 0 + \frac{\sqrt{2}}{2} \cdot 0 \end{array}]$

Step 3.3.2.6.2

Simplify

$R_{4}$ .

$[\begin{array}{c} 1 & - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.7

Perform the row operation

$R_{1} = R_{1} + \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Step 3.3.2.7.1

Perform the row operation

$R_{1} = R_{1} + \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 + \frac{\sqrt{2}}{2} \cdot 0 & - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 1 & 0 + \frac{\sqrt{2}}{2} \sqrt{2} & \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 1 & 0 + \frac{\sqrt{2}}{2} \cdot 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.7.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & \sqrt{2} & 0 \\ 0 & 1 & \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.3

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

$x_{1} + x_{3} + \sqrt{2} x_{4} = 0$

$x_{2} + \sqrt{2} x_{3} + x_{4} = 0$

$0 = 0$

Step 3.3.4

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

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

Step 3.3.5

Write the solution as a linear combination of vectors.

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

Step 3.3.6

Write as a solution set.

${x_{3} [\begin{array}{c} - 1 \\ - \sqrt{2} \\ 1 \\ 0 \end{array}] + x_{4} [\begin{array}{c} - \sqrt{2} \\ - 1 \\ 0 \\ 1 \end{array}] | x_{3}, x_{4} \in R}$

Step 3.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} - 1 \\ - \sqrt{2} \\ 1 \\ 0 \end{array}], [\begin{array}{c} - \sqrt{2} \\ - 1 \\ 0 \\ 1 \end{array}]}$

Step 4

Find the eigenvector using the eigenvalue

$λ = - \sqrt{2}$ .

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Step 4.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + \sqrt{2} [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 4.2

Simplify.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply

$\sqrt{2}$ by each element of the matrix.

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2

Simplify each element in the matrix.

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Step 4.2.1.2.1

Multiply

$\sqrt{2}$ by

$1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.2

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.3

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.4

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.5

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.6

Multiply

$\sqrt{2}$ by

$1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.7

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.8

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.9

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.10

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} \cdot 1 & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.11

Multiply

$\sqrt{2}$ by

$1$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & \sqrt{2} \cdot 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.12

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \\ \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.13

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \\ 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.14

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \cdot 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.15

Multiply

$\sqrt{2}$ by

$0$ .

$[\begin{array}{c} 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 \\ 0 & - 1 & 0 & - 1 \\ - 1 & 0 & - 1 & 0 \end{array}] + [\begin{array}{c} \sqrt{2} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & \sqrt{2} \cdot 1 \end{array}]$

Step 4.2.1.2.16

Multiply

$\sqrt{2}$ by

$1$ .

Step 4.2.2

Add the corresponding elements.

$[\begin{array}{c} 0 + \sqrt{2} & 1 + 0 & 0 + 0 & - 1 + 0 \\ 1 + 0 & 0 + \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3

Simplify each element.

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Step 4.2.3.1

Add

$0$ and

$\sqrt{2}$ .

$[\begin{array}{c} \sqrt{2} & 1 + 0 & 0 + 0 & - 1 + 0 \\ 1 + 0 & 0 + \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.2

Add

$1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 + 0 & - 1 + 0 \\ 1 + 0 & 0 + \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.3

Add

$0$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 + 0 \\ 1 + 0 & 0 + \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.4

Add

$- 1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 + 0 & 0 + \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.5

Add

$1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & 0 + \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.6

Add

$0$ and

$\sqrt{2}$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 + 0 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.7

Add

$- 1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 + 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.8

Add

$0$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 + 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.9

Add

$0$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 + 0 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.10

Add

$- 1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & 0 + \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.11

Add

$0$ and

$\sqrt{2}$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 + 0 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.12

Add

$- 1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 \\ - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.13

Add

$- 1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 \\ - 1 & 0 + 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.14

Add

$0$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 \\ - 1 & 0 & - 1 + 0 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.15

Add

$- 1$ and

$0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 \\ - 1 & 0 & - 1 & 0 + \sqrt{2} \end{array}]$

Step 4.2.3.16

Add

$0$ and

$\sqrt{2}$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 \\ 1 & \sqrt{2} & - 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 \\ - 1 & 0 & - 1 & \sqrt{2} \end{array}]$

Step 4.3

Find the null space when

$λ = - \sqrt{2}$ .

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Step 4.3.1

Write as an augmented matrix for

$A x = 0$ .

$[\begin{array}{c} \sqrt{2} & 1 & 0 & - 1 & 0 \\ 1 & \sqrt{2} & - 1 & 0 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & \sqrt{2} & 0 \end{array}]$

Step 4.3.2

Find the reduced row echelon form.

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Step 4.3.2.1

Multiply each element of

$R_{1}$ by

$\frac{1}{\sqrt{2}}$ to make the entry at

$1, 1$ a

$1$ .

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Step 4.3.2.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{\sqrt{2}}$ to make the entry at

$1, 1$ a

$1$ .

$[\begin{array}{c} \frac{\sqrt{2}}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{0}{\sqrt{2}} & \frac{- 1}{\sqrt{2}} & \frac{0}{\sqrt{2}} \\ 1 & \sqrt{2} & - 1 & 0 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & \sqrt{2} & 0 \end{array}]$

Step 4.3.2.1.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 1 & \sqrt{2} & - 1 & 0 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & \sqrt{2} & 0 \end{array}]$

Step 4.3.2.2

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Step 4.3.2.2.1

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 1 - 1 & \sqrt{2} - \frac{\sqrt{2}}{2} & - 1 - 0 & 0 + \frac{\sqrt{2}}{2} & 0 - 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & \sqrt{2} & 0 \end{array}]$

Step 4.3.2.2.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ - 1 & 0 & - 1 & \sqrt{2} & 0 \end{array}]$

Step 4.3.2.3

Perform the row operation

$R_{4} = R_{4} + R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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Step 4.3.2.3.1

Perform the row operation

$R_{4} = R_{4} + R_{1}$ to make the entry at

$4, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ - 1 + 1 \cdot 1 & 0 + \frac{\sqrt{2}}{2} & - 1 + 0 & \sqrt{2} - \frac{\sqrt{2}}{2} & 0 + 0 \end{array}]$

Step 4.3.2.3.2

Simplify

$R_{4}$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 4.3.2.4

Multiply each element of

$R_{2}$ by

$\frac{2}{\sqrt{2}}$ to make the entry at

$2, 2$ a

$1$ .

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Step 4.3.2.4.1

Multiply each element of

$R_{2}$ by

$\frac{2}{\sqrt{2}}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ \frac{2}{\sqrt{2}} \cdot 0 & \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{2} & \frac{2}{\sqrt{2}} \cdot - 1 & \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{2} & \frac{2}{\sqrt{2}} \cdot 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 4.3.2.4.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 & - 1 & \sqrt{2} & - 1 & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 4.3.2.5

Perform the row operation

$R_{3} = R_{3} + R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Step 4.3.2.5.1

Perform the row operation

$R_{3} = R_{3} + R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 + 0 & - 1 + 1 \cdot 1 & \sqrt{2} - \sqrt{2} & - 1 + 1 \cdot 1 & 0 + 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 4.3.2.5.2

Simplify

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & - 1 & \frac{\sqrt{2}}{2} & 0 \end{array}]$

Step 4.3.2.6

Perform the row operation

$R_{4} = R_{4} - \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Step 4.3.2.6.1

Perform the row operation

$R_{4} = R_{4} - \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$4, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 - \frac{\sqrt{2}}{2} \cdot 0 & \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 1 & - 1 - \frac{\sqrt{2}}{2} (- \sqrt{2}) & \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 1 & 0 - \frac{\sqrt{2}}{2} \cdot 0 \end{array}]$

Step 4.3.2.6.2

Simplify

$R_{4}$ .

$[\begin{array}{c} 1 & \frac{\sqrt{2}}{2} & 0 & - \frac{\sqrt{2}}{2} & 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.7

Perform the row operation

$R_{1} = R_{1} - \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Step 4.3.2.7.1

Perform the row operation

$R_{1} = R_{1} - \frac{\sqrt{2}}{2} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{\sqrt{2}}{2} \cdot 0 & \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 1 & 0 - \frac{\sqrt{2}}{2} (- \sqrt{2}) & - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 1 & 0 - \frac{\sqrt{2}}{2} \cdot 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.7.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & - \sqrt{2} & 0 \\ 0 & 1 & - \sqrt{2} & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.3

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

$x_{1} + x_{3} - \sqrt{2} x_{4} = 0$

$x_{2} - \sqrt{2} x_{3} + x_{4} = 0$

$0 = 0$

Step 4.3.4

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

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

Step 4.3.5

Write the solution as a linear combination of vectors.

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

Step 4.3.6

Write as a solution set.

${x_{3} [\begin{array}{c} - 1 \\ \sqrt{2} \\ 1 \\ 0 \end{array}] + x_{4} [\begin{array}{c} \sqrt{2} \\ - 1 \\ 0 \\ 1 \end{array}] | x_{3}, x_{4} \in R}$

Step 4.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} - 1 \\ \sqrt{2} \\ 1 \\ 0 \end{array}], [\begin{array}{c} \sqrt{2} \\ - 1 \\ 0 \\ 1 \end{array}]}$

Step 5

The eigenspace of

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

${[\begin{array}{c} - 1 \\ - \sqrt{2} \\ 1 \\ 0 \end{array}], [\begin{array}{c} - \sqrt{2} \\ - 1 \\ 0 \\ 1 \end{array}], [\begin{array}{c} - 1 \\ \sqrt{2} \\ 1 \\ 0 \end{array}], [\begin{array}{c} \sqrt{2} \\ - 1 \\ 0 \\ 1 \end{array}]}$