Algebra Examples

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

Step 1

Find the eigenvalues.

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Step 1.1

Set up the formula to find the characteristic equation

p (λ)

$p (λ)$ .

p (λ) = determinant (A - λ I_{3})

$p (λ) = determinant (A - λ I_{3})$

Step 1.2

The identity matrix or unit matrix of size

3

$3$ is the

3 \times 3

$3 \times 3$ square matrix with ones on the main diagonal and zeros elsewhere.

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

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

Step 1.3

Substitute the known values into

p (λ) = determinant (A - λ I_{3})

$p (λ) = determinant (A - λ I_{3})$ .

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Step 1.3.1

Substitute

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$ for

A

$A$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - λ I_{3})

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - λ I_{3})$

Step 1.3.2

Substitute

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

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

I_{3}

$I_{3}$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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

$- λ$ by each element of the matrix.

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply

- 1

$- 1$ by

1

$1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

Step 1.4.1.2.3

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

Step 1.4.1.2.4

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 1.4.1.2.4.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

Step 1.4.1.2.5

Multiply

- 1

$- 1$ by

1

$1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 1.4.1.2.6.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

Step 1.4.1.2.7

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 1.4.1.2.7.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 1.4.1.2.8.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.9

Multiply

- 1

$- 1$ by

1

$1$ .

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])

$p (λ) = determinant ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 1.4.2

Add the corresponding elements.

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 + 0 & 6 + 0 \\ 8 + 0 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 + 0 & 6 + 0 \\ 8 + 0 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]$

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Add

4

$4$ and

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 + 0 \\ 8 + 0 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 + 0 \\ 8 + 0 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]$

Step 1.4.3.2

Add

6

$6$ and

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 + 0 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 + 0 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]$

Step 1.4.3.3

Add

8

$8$ and

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]$

Step 1.4.3.4

Add

12

$12$ and

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 + 0 & 2 + 0 & 0 - λ \end{array}]$

Step 1.4.3.5

Add

1

$1$ and

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 + 0 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 + 0 & 0 - λ \end{array}]$

Step 1.4.3.6

Add

2

$2$ and

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & 0 - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & 0 - λ \end{array}]$

Step 1.4.3.7

Subtract

λ

$λ$ from

0

$0$ .

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & - λ \end{array}]$

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & - λ \end{array}]$

p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & - λ \end{array}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 4 & 6 \\ 8 & 10 - λ & 12 \\ 1 & 2 & - λ \end{array}]$

Step 1.5

Find the determinant.

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Step 1.5.1

Choose the row or column with the most

0

$0$ elements. If there are no

0

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

1

$1$ by its cofactor and add.

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Step 1.5.1.1

Consider the corresponding sign chart.

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

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

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}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

| \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} |

$| \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} |$

Step 1.5.1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

(2 - λ) | \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} |

$(2 - λ) | \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} |$

Step 1.5.1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

| \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} |

$| \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} |$

Step 1.5.1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} |

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

Step 1.5.1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

| \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$| \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.1.9

Add the terms together.

p (λ) = (2 - λ) | \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} | - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) | \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} | - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) | \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} | - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) | \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} | - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2

Evaluate

| \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} |

$| \begin{array}{c} 10 - λ & 12 \\ 2 & - λ \end{array} |$ .

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Step 1.5.2.1

The determinant of a

2 \times 2

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

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

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

p (λ) = (2 - λ) ((10 - λ) (- λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) ((10 - λ) (- λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2

Simplify the determinant.

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Step 1.5.2.2.1

Simplify each term.

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Step 1.5.2.2.1.1

Apply the distributive property.

p (λ) = (2 - λ) (10 (- λ) - λ (- λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (10 (- λ) - λ (- λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.2

Multiply

- 1

$- 1$ by

10

$10$ .

p (λ) = (2 - λ) (- 10 λ - λ (- λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ - λ (- λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.3

Rewrite using the commutative property of multiplication.

p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 λ \cdot λ - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 λ \cdot λ - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.4

Simplify each term.

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Step 1.5.2.2.1.4.1

Multiply

λ

$λ$ by

λ

$λ$ by adding the exponents.

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Step 1.5.2.2.1.4.1.1

Move

λ

$λ$ .

p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 (λ \cdot λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 (λ \cdot λ) - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.4.1.2

Multiply

λ

$λ$ by

λ

$λ$ .

p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ - 1 \cdot - 1 λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.4.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

p (λ) = (2 - λ) (- 10 λ + 1 λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ + 1 λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.4.3

Multiply

λ^{2}

$λ^{2}$ by

1

$1$ .

p (λ) = (2 - λ) (- 10 λ + λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ + λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (- 10 λ + λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ + λ^{2} - 2 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.1.5

Multiply

- 2

$- 2$ by

12

$12$ .

p (λ) = (2 - λ) (- 10 λ + λ^{2} - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ + λ^{2} - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (- 10 λ + λ^{2} - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (- 10 λ + λ^{2} - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.2.2.2

Reorder

- 10 λ

$- 10 λ$ and

λ^{2}

$λ^{2}$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 | \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} | + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.3

Evaluate

| \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} |

$| \begin{array}{c} 8 & 12 \\ 1 & - λ \end{array} |$ .

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Step 1.5.3.1

The determinant of a

2 \times 2

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

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

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

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (8 (- λ) - 1 \cdot 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (8 (- λ) - 1 \cdot 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.3.2

Simplify each term.

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Step 1.5.3.2.1

Multiply

- 1

$- 1$ by

8

$8$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 1 \cdot 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 1 \cdot 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.3.2.2

Multiply

- 1

$- 1$ by

12

$12$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 | \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$

Step 1.5.4

Evaluate

| \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |

$| \begin{array}{c} 8 & 10 - λ \\ 1 & 2 \end{array} |$ .

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Step 1.5.4.1

The determinant of a

2 \times 2

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

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

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

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (8 \cdot 2 - (10 - λ))

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (8 \cdot 2 - (10 - λ))$

Step 1.5.4.2

Simplify the determinant.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Multiply

8

$8$ by

2

$2$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - (10 - λ))

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - (10 - λ))$

Step 1.5.4.2.1.2

Apply the distributive property.

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 1 \cdot 10 - - λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 1 \cdot 10 - - λ)$

Step 1.5.4.2.1.3

Multiply

- 1

$- 1$ by

10

$10$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 - - λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 - - λ)$

Step 1.5.4.2.1.4

Multiply

- - λ

$- - λ$ .

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Step 1.5.4.2.1.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + 1 λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + 1 λ)$

Step 1.5.4.2.1.4.2

Multiply

λ

$λ$ by

1

$1$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + λ)$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + λ)$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (16 - 10 + λ)$

Step 1.5.4.2.2

Subtract

10

$10$ from

16

$16$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (6 + λ)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (6 + λ)$

Step 1.5.4.2.3

Reorder

6

$6$ and

λ

$λ$ .

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (λ + 6)$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (λ + 6)$

p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = (2 - λ) (λ^{2} - 10 λ - 24) - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5

Simplify the determinant.

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Step 1.5.5.1

Simplify each term.

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Step 1.5.5.1.1

Expand

(2 - λ) (λ^{2} - 10 λ - 24)

$(2 - λ) (λ^{2} - 10 λ - 24)$ by multiplying each term in the first expression by each term in the second expression.

p (λ) = 2 λ^{2} + 2 (- 10 λ) + 2 \cdot - 24 - λ \cdot λ^{2} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} + 2 (- 10 λ) + 2 \cdot - 24 - λ \cdot λ^{2} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2

Simplify each term.

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Step 1.5.5.1.2.1

Multiply

- 10

$- 10$ by

2

$2$ .

p (λ) = 2 λ^{2} - 20 λ + 2 \cdot - 24 - λ \cdot λ^{2} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ + 2 \cdot - 24 - λ \cdot λ^{2} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.2

Multiply

2

$2$ by

- 24

$- 24$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - λ \cdot λ^{2} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ \cdot λ^{2} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.3

Multiply

λ

$λ$ by

λ^{2}

$λ^{2}$ by adding the exponents.

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Step 1.5.5.1.2.3.1

Move

λ^{2}

$λ^{2}$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - (λ^{2} λ) - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - (λ^{2} λ) - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.3.2

Multiply

λ^{2}

$λ^{2}$ by

λ

$λ$ .

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Step 1.5.5.1.2.3.2.1

Raise

λ

$λ$ to the power of

1

$1$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - (λ^{2} λ^{1}) - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - (λ^{2} λ^{1}) - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.3.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

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

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{2 + 1} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{2 + 1} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{2 + 1} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{2 + 1} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.3.3

Add

2

$2$ and

1

$1$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - λ (- 10 λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.4

Rewrite using the commutative property of multiplication.

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 λ \cdot λ - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 λ \cdot λ - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.5

Multiply

λ

$λ$ by

λ

$λ$ by adding the exponents.

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Step 1.5.5.1.2.5.1

Move

λ

$λ$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 (λ \cdot λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 (λ \cdot λ) - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.5.2

Multiply

λ

$λ$ by

λ

$λ$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 λ^{2} - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 λ^{2} - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 λ^{2} - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} - 1 \cdot - 10 λ^{2} - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.6

Multiply

- 1

$- 1$ by

- 10

$- 10$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} + 10 λ^{2} - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} + 10 λ^{2} - λ \cdot - 24 - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.2.7

Multiply

- 24

$- 24$ by

- 1

$- 1$ .

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} + 10 λ^{2} + 24 λ - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} + 10 λ^{2} + 24 λ - 4 (- 8 λ - 12) + 6 (λ + 6)$

p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} + 10 λ^{2} + 24 λ - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 2 λ^{2} - 20 λ - 48 - λ^{3} + 10 λ^{2} + 24 λ - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.3

Add

2 λ^{2}

$2 λ^{2}$ and

10 λ^{2}

$10 λ^{2}$ .

p (λ) = 12 λ^{2} - 20 λ - 48 - λ^{3} + 24 λ - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 12 λ^{2} - 20 λ - 48 - λ^{3} + 24 λ - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.4

Add

- 20 λ

$- 20 λ$ and

24 λ

$24 λ$ .

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} - 4 (- 8 λ - 12) + 6 (λ + 6)

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} - 4 (- 8 λ - 12) + 6 (λ + 6)$

Step 1.5.5.1.5

Apply the distributive property.

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} - 4 (- 8 λ) - 4 \cdot - 12 + 6 (λ + 6)

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} - 4 (- 8 λ) - 4 \cdot - 12 + 6 (λ + 6)$

Step 1.5.5.1.6

Multiply

- 8

$- 8$ by

- 4

$- 4$ .

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ - 4 \cdot - 12 + 6 (λ + 6)

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ - 4 \cdot - 12 + 6 (λ + 6)$

Step 1.5.5.1.7

Multiply

- 4

$- 4$ by

- 12

$- 12$ .

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 (λ + 6)

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 (λ + 6)$

Step 1.5.5.1.8

Apply the distributive property.

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 6 \cdot 6

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 6 \cdot 6$

Step 1.5.5.1.9

Multiply

6

$6$ by

6

$6$ .

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 36

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 36$

p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 36

$p (λ) = 12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 36$

Step 1.5.5.2

Combine the opposite terms in

12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 36

$12 λ^{2} + 4 λ - 48 - λ^{3} + 32 λ + 48 + 6 λ + 36$ .

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Step 1.5.5.2.1

Add

- 48

$- 48$ and

48

$48$ .

p (λ) = 12 λ^{2} + 4 λ - λ^{3} + 32 λ + 0 + 6 λ + 36

$p (λ) = 12 λ^{2} + 4 λ - λ^{3} + 32 λ + 0 + 6 λ + 36$

Step 1.5.5.2.2

Add

12 λ^{2} + 4 λ - λ^{3} + 32 λ

$12 λ^{2} + 4 λ - λ^{3} + 32 λ$ and

0

$0$ .

p (λ) = 12 λ^{2} + 4 λ - λ^{3} + 32 λ + 6 λ + 36

$p (λ) = 12 λ^{2} + 4 λ - λ^{3} + 32 λ + 6 λ + 36$

p (λ) = 12 λ^{2} + 4 λ - λ^{3} + 32 λ + 6 λ + 36

$p (λ) = 12 λ^{2} + 4 λ - λ^{3} + 32 λ + 6 λ + 36$

Step 1.5.5.3

Add

4 λ

$4 λ$ and

32 λ

$32 λ$ .

p (λ) = 12 λ^{2} - λ^{3} + 36 λ + 6 λ + 36

$p (λ) = 12 λ^{2} - λ^{3} + 36 λ + 6 λ + 36$

Step 1.5.5.4

Add

36 λ

$36 λ$ and

6 λ

$6 λ$ .

p (λ) = 12 λ^{2} - λ^{3} + 42 λ + 36

$p (λ) = 12 λ^{2} - λ^{3} + 42 λ + 36$

Step 1.5.5.5

Reorder

12 λ^{2}

$12 λ^{2}$ and

- λ^{3}

$- λ^{3}$ .

p (λ) = - λ^{3} + 12 λ^{2} + 42 λ + 36

$p (λ) = - λ^{3} + 12 λ^{2} + 42 λ + 36$

p (λ) = - λ^{3} + 12 λ^{2} + 42 λ + 36

$p (λ) = - λ^{3} + 12 λ^{2} + 42 λ + 36$

p (λ) = - λ^{3} + 12 λ^{2} + 42 λ + 36

$p (λ) = - λ^{3} + 12 λ^{2} + 42 λ + 36$

Step 1.6

Set the characteristic polynomial equal to

0

$0$ to find the eigenvalues

λ

$λ$ .

- λ^{3} + 12 λ^{2} + 42 λ + 36 = 0

$- λ^{3} + 12 λ^{2} + 42 λ + 36 = 0$

Step 1.7

Solve for

λ

$λ$ .

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Step 1.7.1

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

λ \approx 14.96690066

$λ \approx 14.96690066$

λ \approx 14.96690066

$λ \approx 14.96690066$

λ \approx 14.96690066

$λ \approx 14.96690066$

Step 2

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

N

$N$ is the null space and

I

$I$ is the identity matrix.

ε_{A} = N (A - λ I_{3})

$ε_{A} = N (A - λ I_{3})$

Step 3

Find the eigenvector using the eigenvalue

λ = 14.96690066

$λ = 14.96690066$ .

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Step 3.1

Substitute the known values into the formula.

N ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - 14.96690066 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])

$N ([\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] - 14.96690066 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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

- 14.96690066

$- 14.96690066$ by each element of the matrix.

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply

- 14.96690066

$- 14.96690066$ by

1

$1$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply

- 14.96690066

$- 14.96690066$ by

0

$0$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply

- 14.96690066

$- 14.96690066$ by

0

$0$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply

- 14.96690066

$- 14.96690066$ by

0

$0$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 \cdot 1 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.5

Multiply

- 14.96690066

$- 14.96690066$ by

1

$1$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & - 14.96690066 \cdot 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.6

Multiply

- 14.96690066

$- 14.96690066$ by

0

$0$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ - 14.96690066 \cdot 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.7

Multiply

- 14.96690066

$- 14.96690066$ by

0

$0$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & - 14.96690066 \cdot 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.8

Multiply

- 14.96690066

$- 14.96690066$ by

0

$0$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \cdot 1 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \cdot 1 \end{array}]$

Step 3.2.1.2.9

Multiply

- 14.96690066

$- 14.96690066$ by

1

$1$ .

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \end{array}]$

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \end{array}]$

[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \end{array}]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}] + [\begin{array}{c} - 14.96690066 & 0 & 0 \\ 0 & - 14.96690066 & 0 \\ 0 & 0 & - 14.96690066 \end{array}]$

Step 3.2.2

Add the corresponding elements.

[\begin{array}{c} 2 - 14.96690066 & 4 + 0 & 6 + 0 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} 2 - 14.96690066 & 4 + 0 & 6 + 0 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3

Simplify each element.

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Step 3.2.3.1

Subtract

14.96690066

$14.96690066$ from

2

$2$ .

[\begin{array}{c} - 12.96690066 & 4 + 0 & 6 + 0 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 + 0 & 6 + 0 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.2

Add

4

$4$ and

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 + 0 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 + 0 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.3

Add

6

$6$ and

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 + 0 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.4

Add

8

$8$ and

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & 10 - 14.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.5

Subtract

14.96690066

$14.96690066$ from

10

$10$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 + 0 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.6

Add

12

$12$ and

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 + 0 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.7

Add

1

$1$ and

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 + 0 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 + 0 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.8

Add

2

$2$ and

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & 0 - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & 0 - 14.96690066 \end{array}]$

Step 3.2.3.9

Subtract

14.96690066

$14.96690066$ from

0

$0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & - 14.96690066 \end{array}]$

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & - 14.96690066 \end{array}]$

[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & - 14.96690066 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 \\ 8 & - 4.96690066 & 12 \\ 1 & 2 & - 14.96690066 \end{array}]$

Step 3.3

Find the null space when

λ = 14.96690066

$λ = 14.96690066$ .

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Step 3.3.1

Write as an augmented matrix for

A x = 0

$A x = 0$ .

[\begin{array}{c} - 12.96690066 & 4 & 6 & 0 \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} - 12.96690066 & 4 & 6 & 0 \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

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}

$R_{1}$ by

\frac{1}{- 12.96690066}

$\frac{1}{- 12.96690066}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Step 3.3.2.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{- 12.96690066}

$\frac{1}{- 12.96690066}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{- 12.96690066}{- 12.96690066} & \frac{4}{- 12.96690066} & \frac{6}{- 12.96690066} & \frac{0}{- 12.96690066} \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} \frac{- 12.96690066}{- 12.96690066} & \frac{4}{- 12.96690066} & \frac{6}{- 12.96690066} & \frac{0}{- 12.96690066} \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

Step 3.3.2.1.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 8 & - 4.96690066 & 12 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

Step 3.3.2.2

Perform the row operation

R_{2} = R_{2} - 8 R_{1}

$R_{2} = R_{2} - 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 3.3.2.2.1

Perform the row operation

R_{2} = R_{2} - 8 R_{1}

$R_{2} = R_{2} - 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 8 - 8 \cdot 1 & - 4.96690066 - 8 \cdot - 0.30847772 & 12 - 8 \cdot - 0.46271658 & 0 - 8 \cdot 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 8 - 8 \cdot 1 & - 4.96690066 - 8 \cdot - 0.30847772 & 12 - 8 \cdot - 0.46271658 & 0 - 8 \cdot 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

Step 3.3.2.2.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 1 & 2 & - 14.96690066 & 0 \end{array}]$

Step 3.3.2.3

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Step 3.3.2.3.1

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 1 - 1 & 2 + 0.30847772 & - 14.96690066 + 0.46271658 & 0 - 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 1 - 1 & 2 + 0.30847772 & - 14.96690066 + 0.46271658 & 0 - 0 \end{array}]$

Step 3.3.2.3.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & - 2.49907888 & 15.70173268 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]$

Step 3.3.2.4

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{- 2.49907888}

$\frac{1}{- 2.49907888}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 3.3.2.4.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{- 2.49907888}

$\frac{1}{- 2.49907888}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ \frac{0}{- 2.49907888} & \frac{- 2.49907888}{- 2.49907888} & \frac{15.70173268}{- 2.49907888} & \frac{0}{- 2.49907888} \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ \frac{0}{- 2.49907888} & \frac{- 2.49907888}{- 2.49907888} & \frac{15.70173268}{- 2.49907888} & \frac{0}{- 2.49907888} \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]$

Step 3.3.2.4.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 2.30847772 & - 14.50418408 & 0 \end{array}]$

Step 3.3.2.5

Perform the row operation

R_{3} = R_{3} - 2.30847772 R_{2}

$R_{3} = R_{3} - 2.30847772 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Step 3.3.2.5.1

Perform the row operation

R_{3} = R_{3} - 2.30847772 R_{2}

$R_{3} = R_{3} - 2.30847772 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 - 2.30847772 \cdot 0 & 2.30847772 - 2.30847772 \cdot 1 & - 14.50418408 - 2.30847772 \cdot - 6.28300803 & 0 - 2.30847772 \cdot 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 - 2.30847772 \cdot 0 & 2.30847772 - 2.30847772 \cdot 1 & - 14.50418408 - 2.30847772 \cdot - 6.28300803 & 0 - 2.30847772 \cdot 0 \end{array}]$

Step 3.3.2.5.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.6

Multiply each element of

R_{3}

$R_{3}$ by

\frac{1}{2.7504016 \cdot 10^{- 12}}

$\frac{1}{2.7504016 \cdot 10^{- 12}}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Step 3.3.2.6.1

Multiply each element of

R_{3}

$R_{3}$ by

\frac{1}{2.7504016 \cdot 10^{- 12}}

$\frac{1}{2.7504016 \cdot 10^{- 12}}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ \frac{0}{2.7504016 \cdot 10^{- 12}} & \frac{0}{2.7504016 \cdot 10^{- 12}} & \frac{0}{2.7504016 \cdot 10^{- 12}} & \frac{0}{2.7504016 \cdot 10^{- 12}} \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ \frac{0}{2.7504016 \cdot 10^{- 12}} & \frac{0}{2.7504016 \cdot 10^{- 12}} & \frac{0}{2.7504016 \cdot 10^{- 12}} & \frac{0}{2.7504016 \cdot 10^{- 12}} \end{array}]$

Step 3.3.2.6.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & - 6.28300803 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.7

Perform the row operation

R_{2} = R_{2} + 6.28300803 R_{3}

$R_{2} = R_{2} + 6.28300803 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Step 3.3.2.7.1

Perform the row operation

R_{2} = R_{2} + 6.28300803 R_{3}

$R_{2} = R_{2} + 6.28300803 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 + 6.28300803 \cdot 0 & 1 + 6.28300803 \cdot 0 & - 6.28300803 + 6.28300803 \cdot 1 & 0 + 6.28300803 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 + 6.28300803 \cdot 0 & 1 + 6.28300803 \cdot 0 & - 6.28300803 + 6.28300803 \cdot 1 & 0 + 6.28300803 \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.7.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & - 0.46271658 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.8

Perform the row operation

R_{1} = R_{1} + 0.46271658 R_{3}

$R_{1} = R_{1} + 0.46271658 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Step 3.3.2.8.1

Perform the row operation

R_{1} = R_{1} + 0.46271658 R_{3}

$R_{1} = R_{1} + 0.46271658 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

[\begin{array}{c} 1 + 0.46271658 \cdot 0 & - 0.30847772 + 0.46271658 \cdot 0 & - 0.46271658 + 0.46271658 \cdot 1 & 0 + 0.46271658 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 + 0.46271658 \cdot 0 & - 0.30847772 + 0.46271658 \cdot 0 & - 0.46271658 + 0.46271658 \cdot 1 & 0 + 0.46271658 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.8.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & - 0.30847772 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

[\begin{array}{c} 1 & - 0.30847772 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 & - 0.30847772 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.9

Perform the row operation

R_{1} = R_{1} + 0.30847772 R_{2}

$R_{1} = R_{1} + 0.30847772 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 3.3.2.9.1

Perform the row operation

R_{1} = R_{1} + 0.30847772 R_{2}

$R_{1} = R_{1} + 0.30847772 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 + 0.30847772 \cdot 0 & - 0.30847772 + 0.30847772 \cdot 1 & 0 + 0.30847772 \cdot 0 & 0 + 0.30847772 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]

$[\begin{array}{c} 1 + 0.30847772 \cdot 0 & - 0.30847772 + 0.30847772 \cdot 1 & 0 + 0.30847772 \cdot 0 & 0 + 0.30847772 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 3.3.2.9.2

Simplify

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Step 3.3.3

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

x = 0

$x = 0$

y = 0

$y = 0$

z = 0

$z = 0$

Step 3.3.4

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

[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 3.3.5

Write as a solution set.

{[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}

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

{[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}

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

{[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}

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

Step 4

The eigenspace of

A

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

{[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}

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

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