Linear Algebra Examples

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

Step 1

Find the eigenvalues.

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Step 1.1

Set up the formula to find the characteristic equation $p (λ)$ .

$p (λ) = determinant (A - λ I_{3})$

Step 1.2

The identity matrix or unit matrix of size $3$ is the $3 \times 3$ square matrix with ones on the main diagonal and zeros elsewhere.

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

Step 1.3

Substitute the known values into $p (λ) = determinant (A - λ I_{3})$ .

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Step 1.3.1

Substitute $[\begin{array}{c} 1 & 2 & - 1 \\ 1 & 0 & 1 \\ - 2 & - 2 & 0 \end{array}]$ for $A$ .

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

Step 1.3.2

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

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

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 $- λ$ by each element of the matrix.

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

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply $- 1$ by $1$ .

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

Step 1.4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply $0$ by $- 1$ .

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

Step 1.4.1.2.2.2

Multiply $0$ by $λ$ .

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

Step 1.4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply $0$ by $- 1$ .

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

Step 1.4.1.2.3.2

Multiply $0$ by $λ$ .

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

Step 1.4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.4.1

Multiply $0$ by $- 1$ .

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

Step 1.4.1.2.4.2

Multiply $0$ by $λ$ .

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

Step 1.4.1.2.5

Multiply $- 1$ by $1$ .

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

Step 1.4.1.2.6

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.6.1

Multiply $0$ by $- 1$ .

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

Step 1.4.1.2.6.2

Multiply $0$ by $λ$ .

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

Step 1.4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.7.1

Multiply $0$ by $- 1$ .

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

Step 1.4.1.2.7.2

Multiply $0$ by $λ$ .

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

Step 1.4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.8.1

Multiply $0$ by $- 1$ .

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

Step 1.4.1.2.8.2

Multiply $0$ by $λ$ .

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

Step 1.4.1.2.9

Multiply $- 1$ by $1$ .

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

Step 1.4.2

Add the corresponding elements.

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

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Add $2$ and $0$ .

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

Step 1.4.3.2

Add $- 1$ and $0$ .

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

Step 1.4.3.3

Add $1$ and $0$ .

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

Step 1.4.3.4

Subtract $λ$ from $0$ .

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

Step 1.4.3.5

Add $1$ and $0$ .

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

Step 1.4.3.6

Add $- 2$ and $0$ .

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

Step 1.4.3.7

Add $- 2$ and $0$ .

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

Step 1.4.3.8

Subtract $λ$ from $0$ .

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

Step 1.5

Find the determinant.

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Step 1.5.1

Choose the row or column with the most $0$ 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.

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

Step 1.5.1.4

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

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

Step 1.5.1.5

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

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

Step 1.5.1.6

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

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

Step 1.5.1.7

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

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

Step 1.5.1.8

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

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

Step 1.5.1.9

Add the terms together.

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Simplify each term.

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Step 1.5.2.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.2.2.2

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

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Step 1.5.2.2.2.1

Move $λ$ .

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

Step 1.5.2.2.2.2

Multiply $λ$ by $λ$ .

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

Step 1.5.2.2.3

Multiply $- 1$ by $- 1$ .

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

Step 1.5.2.2.4

Multiply $λ^{2}$ by $1$ .

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

Step 1.5.2.2.5

Multiply $- (- 2 \cdot 1)$ .

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Step 1.5.2.2.5.1

Multiply $- 2$ by $1$ .

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

Step 1.5.2.2.5.2

Multiply $- 1$ by $- 2$ .

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Simplify each term.

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Step 1.5.3.2.1

Multiply $- λ$ by $1$ .

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

Step 1.5.3.2.2

Multiply $- (- 2 \cdot 1)$ .

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Step 1.5.3.2.2.1

Multiply $- 2$ by $1$ .

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

Step 1.5.3.2.2.2

Multiply $- 1$ by $- 2$ .

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

Step 1.5.4

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

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Step 1.5.4.1

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

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

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 $- 2$ by $1$ .

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

Step 1.5.4.2.1.2

Multiply $- 1$ by $- 2$ .

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

Step 1.5.4.2.1.3

Multiply $2$ by $- 1$ .

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

Step 1.5.4.2.2

Reorder $- 2$ and $- 2 λ$ .

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

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

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Step 1.5.5.1.1.1

Apply the distributive property.

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

Step 1.5.5.1.1.2

Apply the distributive property.

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

Step 1.5.5.1.1.3

Apply the distributive property.

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

Step 1.5.5.1.2

Simplify each term.

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Step 1.5.5.1.2.1

Multiply $λ^{2}$ by $1$ .

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

Step 1.5.5.1.2.2

Multiply $2$ by $1$ .

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

Step 1.5.5.1.2.3

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

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Step 1.5.5.1.2.3.1

Move $λ^{2}$ .

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

Step 1.5.5.1.2.3.2

Multiply $λ^{2}$ by $λ$ .

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Step 1.5.5.1.2.3.2.1

Raise $λ$ to the power of $1$ .

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

Step 1.5.5.1.2.3.2.2

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

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

Step 1.5.5.1.2.3.3

Add $2$ and $1$ .

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

Step 1.5.5.1.2.4

Multiply $2$ by $- 1$ .

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

Step 1.5.5.1.3

Apply the distributive property.

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

Step 1.5.5.1.4

Multiply $- 1$ by $- 2$ .

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

Step 1.5.5.1.5

Multiply $- 2$ by $2$ .

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

Step 1.5.5.1.6

Apply the distributive property.

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

Step 1.5.5.1.7

Multiply $- 2$ by $- 1$ .

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

Step 1.5.5.1.8

Multiply $- 1$ by $- 2$ .

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

Step 1.5.5.2

Combine the opposite terms in $λ^{2} + 2 - λ^{3} - 2 λ + 2 λ - 4 + 2 λ + 2$ .

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Step 1.5.5.2.1

Add $- 2 λ$ and $2 λ$ .

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

Step 1.5.5.2.2

Add $λ^{2} + 2 - λ^{3}$ and $0$ .

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

Step 1.5.5.3

Subtract $4$ from $2$ .

$p (λ) = λ^{2} - λ^{3} - 2 + 2 λ + 2$

Step 1.5.5.4

Combine the opposite terms in $λ^{2} - λ^{3} - 2 + 2 λ + 2$ .

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Step 1.5.5.4.1

Add $- 2$ and $2$ .

$p (λ) = λ^{2} - λ^{3} + 2 λ + 0$

Step 1.5.5.4.2

Add $λ^{2} - λ^{3} + 2 λ$ and $0$ .

$p (λ) = λ^{2} - λ^{3} + 2 λ$

Step 1.5.5.5

Reorder $λ^{2}$ and $- λ^{3}$ .

$p (λ) = - λ^{3} + λ^{2} + 2 λ$

Step 1.6

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

$- λ^{3} + λ^{2} + 2 λ = 0$

Step 1.7

Solve for $λ$ .

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Step 1.7.1

Factor the left side of the equation.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

Factor $- λ$ out of $- λ^{3}$ .

$- λ \cdot λ^{2} + λ^{2} + 2 λ = 0$

Step 1.7.1.1.2

Factor $- λ$ out of $λ^{2}$ .

$- λ \cdot λ^{2} - λ (- λ) + 2 λ = 0$

Step 1.7.1.1.3

Factor $- λ$ out of $2 λ$ .

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

Step 1.7.1.1.4

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

$- λ (λ^{2} - λ) - λ \cdot - 2 = 0$

Step 1.7.1.1.5

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

$- λ (λ^{2} - λ - 2) = 0$

Step 1.7.1.2

Factor.

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Step 1.7.1.2.1

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

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Step 1.7.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.7.1.2.1.2

Write the factored form using these integers.

$- λ ((λ - 2) (λ + 1)) = 0$

Step 1.7.1.2.2

Remove unnecessary parentheses.

$- λ (λ - 2) (λ + 1) = 0$

Step 1.7.2

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

$λ = 0$

$λ - 2 = 0$

$λ + 1 = 0$

Step 1.7.3

Set $λ$ equal to $0$ .

$λ = 0$

Step 1.7.4

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

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Step 1.7.4.1

Set $λ - 2$ equal to $0$ .

$λ - 2 = 0$

Step 1.7.4.2

Add $2$ to both sides of the equation.

$λ = 2$

Step 1.7.5

Set $λ + 1$ equal to $0$ and solve for $λ$ .

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Step 1.7.5.1

Set $λ + 1$ equal to $0$ .

$λ + 1 = 0$

Step 1.7.5.2

Subtract $1$ from both sides of the equation.

$λ = - 1$

Step 1.7.6

The final solution is all the values that make $- λ (λ - 2) (λ + 1) = 0$ true.

$λ = 0, 2, - 1$

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.

$ε_{A} = N (A - λ I_{3})$

Step 3

Find the eigenvector using the eigenvalue $λ = 0$ .

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Step 3.1

Substitute the known values into the formula.

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

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 $0$ by each element of the matrix.

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

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply $0$ by $1$ .

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

Step 3.2.1.2.2

Multiply $0$ by $0$ .

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

Step 3.2.1.2.3

Multiply $0$ by $0$ .

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

Step 3.2.1.2.4

Multiply $0$ by $0$ .

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

Step 3.2.1.2.5

Multiply $0$ by $1$ .

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

Step 3.2.1.2.6

Multiply $0$ by $0$ .

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

Step 3.2.1.2.7

Multiply $0$ by $0$ .

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

Step 3.2.1.2.8

Multiply $0$ by $0$ .

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

Step 3.2.1.2.9

Multiply $0$ by $1$ .

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

Step 3.2.2

Adding any matrix to a null matrix is the matrix itself.

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Step 3.2.2.1

Add the corresponding elements.

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

Step 3.2.2.2

Simplify each element.

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Step 3.2.2.2.1

Add $1$ and $0$ .

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

Step 3.2.2.2.2

Add $2$ and $0$ .

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

Step 3.2.2.2.3

Add $- 1$ and $0$ .

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

Step 3.2.2.2.4

Add $1$ and $0$ .

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

Step 3.2.2.2.5

Add $0$ and $0$ .

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

Step 3.2.2.2.6

Add $1$ and $0$ .

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

Step 3.2.2.2.7

Add $- 2$ and $0$ .

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

Step 3.2.2.2.8

Add $- 2$ and $0$ .

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

Step 3.2.2.2.9

Add $0$ and $0$ .

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

Step 3.3

Find the null space when $λ = 0$ .

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Step 3.3.1

Write as an augmented matrix for $A x = 0$ .

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

Step 3.3.2

Find the reduced row echelon form.

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Step 3.3.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.3.2.1.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Step 3.3.2.1.2

Simplify $R_{2}$ .

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

Step 3.3.2.2

Perform the row operation $R_{3} = R_{3} + 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.3.2.2.1

Perform the row operation $R_{3} = R_{3} + 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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

Step 3.3.2.2.2

Simplify $R_{3}$ .

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

Step 3.3.2.3

Multiply each element of $R_{2}$ by $- \frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

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Step 3.3.2.3.1

Multiply each element of $R_{2}$ by $- \frac{1}{2}$ to make the entry at $2, 2$ a $1$ .

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

Step 3.3.2.3.2

Simplify $R_{2}$ .

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

Step 3.3.2.4

Perform the row operation $R_{3} = R_{3} - 2 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 3.3.2.4.1

Perform the row operation $R_{3} = R_{3} - 2 R_{2}$ to make the entry at $3, 2$ a $0$ .

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

Step 3.3.2.4.2

Simplify $R_{3}$ .

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

Step 3.3.2.5

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.2.5.1

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Step 3.3.2.5.2

Simplify $R_{1}$ .

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

Step 3.3.3

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

$x + z = 0$

$y - z = 0$

$0 = 0$

Step 3.3.4

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

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

Step 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} - 1 \\ 1 \\ 1 \end{array}]$

Step 3.3.6

Write as a solution set.

${z [\begin{array}{c} - 1 \\ 1 \\ 1 \end{array}] | z \in R}$

Step 3.3.7

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

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

Step 4

Find the eigenvector using the eigenvalue $λ = 2$ .

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Step 4.1

Substitute the known values into the formula.

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

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 $- 2$ by each element of the matrix.

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

Step 4.2.1.2

Simplify each element in the matrix.

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Step 4.2.1.2.1

Multiply $- 2$ by $1$ .

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

Step 4.2.1.2.2

Multiply $- 2$ by $0$ .

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

Step 4.2.1.2.3

Multiply $- 2$ by $0$ .

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

Step 4.2.1.2.4

Multiply $- 2$ by $0$ .

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

Step 4.2.1.2.5

Multiply $- 2$ by $1$ .

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

Step 4.2.1.2.6

Multiply $- 2$ by $0$ .

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

Step 4.2.1.2.7

Multiply $- 2$ by $0$ .

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

Step 4.2.1.2.8

Multiply $- 2$ by $0$ .

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

Step 4.2.1.2.9

Multiply $- 2$ by $1$ .

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

Step 4.2.2

Add the corresponding elements.

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

Step 4.2.3

Simplify each element.

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Step 4.2.3.1

Subtract $2$ from $1$ .

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

Step 4.2.3.2

Add $2$ and $0$ .

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

Step 4.2.3.3

Add $- 1$ and $0$ .

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

Step 4.2.3.4

Add $1$ and $0$ .

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

Step 4.2.3.5

Subtract $2$ from $0$ .

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

Step 4.2.3.6

Add $1$ and $0$ .

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

Step 4.2.3.7

Add $- 2$ and $0$ .

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

Step 4.2.3.8

Add $- 2$ and $0$ .

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

Step 4.2.3.9

Subtract $2$ from $0$ .

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

Step 4.3

Find the null space when $λ = 2$ .

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Step 4.3.1

Write as an augmented matrix for $A x = 0$ .

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

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 $- 1$ 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 $- 1$ to make the entry at $1, 1$ a $1$ .

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

Step 4.3.2.1.2

Simplify $R_{1}$ .

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

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 $0$ .

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

Step 4.3.2.2.2

Simplify $R_{2}$ .

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

Step 4.3.2.3

Perform the row operation $R_{3} = R_{3} + 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 4.3.2.3.1

Perform the row operation $R_{3} = R_{3} + 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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

Step 4.3.2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 6 & 0 & 0 \end{array}]$

Step 4.3.2.4

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ 0 & - 6 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.5

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Step 4.3.2.5.1

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 1 & 0 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.2.5.2

Simplify $R_{2}$ .

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

Step 4.3.2.6

Perform the row operation $R_{1} = R_{1} + 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.3.2.6.1

Perform the row operation $R_{1} = R_{1} + 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Step 4.3.2.6.2

Simplify $R_{1}$ .

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

Step 4.3.3

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

$x + z = 0$

$y = 0$

$0 = 0$

Step 4.3.4

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

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

Step 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} - 1 \\ 0 \\ 1 \end{array}]$

Step 4.3.6

Write as a solution set.

${z [\begin{array}{c} - 1 \\ 0 \\ 1 \end{array}] | z \in R}$

Step 4.3.7

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

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

Step 5

Find the eigenvector using the eigenvalue $λ = - 1$ .

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Step 5.1

Substitute the known values into the formula.

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

Step 5.2

Simplify.

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Step 5.2.1

Add the corresponding elements.

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

Step 5.2.2

Simplify each element.

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Step 5.2.2.1

Add $1$ and $1$ .

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

Step 5.2.2.2

Add $2$ and $0$ .

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

Step 5.2.2.3

Add $- 1$ and $0$ .

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

Step 5.2.2.4

Add $1$ and $0$ .

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

Step 5.2.2.5

Add $0$ and $1$ .

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

Step 5.2.2.6

Add $1$ and $0$ .

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

Step 5.2.2.7

Add $- 2$ and $0$ .

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

Step 5.2.2.8

Add $- 2$ and $0$ .

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

Step 5.2.2.9

Add $0$ and $1$ .

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

Step 5.3

Find the null space when $λ = - 1$ .

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Step 5.3.1

Write as an augmented matrix for $A x = 0$ .

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

Step 5.3.2

Find the reduced row echelon form.

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Step 5.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Step 5.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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

Step 5.3.2.1.2

Simplify $R_{1}$ .

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

Step 5.3.2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 5.3.2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Step 5.3.2.2.2

Simplify $R_{2}$ .

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

Step 5.3.2.3

Perform the row operation $R_{3} = R_{3} + 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 5.3.2.3.1

Perform the row operation $R_{3} = R_{3} + 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & \frac{3}{2} & 0 \\ - 2 + 2 \cdot 1 & - 2 + 2 \cdot 1 & 1 + 2 (- \frac{1}{2}) & 0 + 2 \cdot 0 \end{array}]$

Step 5.3.2.3.2

Simplify $R_{3}$ .

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

Step 5.3.2.4

Multiply each element of $R_{2}$ by $\frac{2}{3}$ to make the entry at $2, 3$ a $1$ .

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Step 5.3.2.4.1

Multiply each element of $R_{2}$ by $\frac{2}{3}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & - \frac{1}{2} & 0 \\ \frac{2}{3} \cdot 0 & \frac{2}{3} \cdot 0 & \frac{2}{3} \cdot \frac{3}{2} & \frac{2}{3} \cdot 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 5.3.2.4.2

Simplify $R_{2}$ .

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

Step 5.3.2.5

Perform the row operation $R_{1} = R_{1} + \frac{1}{2} R_{2}$ to make the entry at $1, 3$ a $0$ .

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Step 5.3.2.5.1

Perform the row operation $R_{1} = R_{1} + \frac{1}{2} R_{2}$ to make the entry at $1, 3$ a $0$ .

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

Step 5.3.2.5.2

Simplify $R_{1}$ .

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

Step 5.3.3

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

$x + y = 0$

$z = 0$

$0 = 0$

Step 5.3.4

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

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

Step 5.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = y [\begin{array}{c} - 1 \\ 1 \\ 0 \end{array}]$

Step 5.3.6

Write as a solution set.

${y [\begin{array}{c} - 1 \\ 1 \\ 0 \end{array}] | y \in R}$

Step 5.3.7

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

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

Step 6

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

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