Linear Algebra Examples

$B = [\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}]$

Step 1

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

$p (λ) = determinant (A - λ I_{3})$

Step 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 3

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

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Step 3.1

Substitute $[\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}]$ for $A$ .

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- λ$ by each element of the matrix.

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply $- λ \cdot 0$ .

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Step 4.1.2.6.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} - 1 & 4 & 3 \\ 1 & 1 & 2 \\ - 1 & 0 & - 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} - 1 - λ & 4 + 0 & 3 + 0 \\ 1 + 0 & 1 - λ & 2 + 0 \\ - 1 + 0 & 0 + 0 & - 1 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - 1 - λ & 4 & 3 + 0 \\ 1 + 0 & 1 - λ & 2 + 0 \\ - 1 + 0 & 0 + 0 & - 1 - λ \end{array}]$

Step 4.3.2

Add $3$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - 1 - λ & 4 & 3 \\ 1 + 0 & 1 - λ & 2 + 0 \\ - 1 + 0 & 0 + 0 & - 1 - λ \end{array}]$

Step 4.3.3

Add $1$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - 1 - λ & 4 & 3 \\ 1 & 1 - λ & 2 + 0 \\ - 1 + 0 & 0 + 0 & - 1 - λ \end{array}]$

Step 4.3.4

Add $2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - 1 - λ & 4 & 3 \\ 1 & 1 - λ & 2 \\ - 1 + 0 & 0 + 0 & - 1 - λ \end{array}]$

Step 4.3.5

Add $- 1$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - 1 - λ & 4 & 3 \\ 1 & 1 - λ & 2 \\ - 1 & 0 + 0 & - 1 - λ \end{array}]$

Step 4.3.6

Add $0$ and $0$ .

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

Step 5

Find the determinant.

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Step 5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $2$ by its cofactor and add.

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Step 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.1.3

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

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

Step 5.1.4

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

$- 4 | \begin{array}{c} 1 & 2 \\ - 1 & - 1 - λ \end{array} |$

Step 5.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

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

Step 5.1.6

Multiply element $a_{22}$ by its cofactor.

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

Step 5.1.7

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

$| \begin{array}{c} - 1 - λ & 3 \\ 1 & 2 \end{array} |$

Step 5.1.8

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

$0 | \begin{array}{c} - 1 - λ & 3 \\ 1 & 2 \end{array} |$

Step 5.1.9

Add the terms together.

$p (λ) = - 4 | \begin{array}{c} 1 & 2 \\ - 1 & - 1 - λ \end{array} | + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0 | \begin{array}{c} - 1 - λ & 3 \\ 1 & 2 \end{array} |$

Step 5.2

Multiply $0$ by $| \begin{array}{c} - 1 - λ & 3 \\ 1 & 2 \end{array} |$ .

$p (λ) = - 4 | \begin{array}{c} 1 & 2 \\ - 1 & - 1 - λ \end{array} | + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0$

Step 5.3

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

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Step 5.3.1

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

$p (λ) = - 4 (1 (- 1 - λ) - (- 1 \cdot 2)) + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0$

Step 5.3.2

Simplify the determinant.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $- 1 - λ$ by $1$ .

$p (λ) = - 4 (- 1 - λ - (- 1 \cdot 2)) + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0$

Step 5.3.2.1.2

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

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Step 5.3.2.1.2.1

Multiply $- 1$ by $2$ .

$p (λ) = - 4 (- 1 - λ - - 2) + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0$

Step 5.3.2.1.2.2

Multiply $- 1$ by $- 2$ .

$p (λ) = - 4 (- 1 - λ + 2) + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0$

Step 5.3.2.2

Add $- 1$ and $2$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) | \begin{array}{c} - 1 - λ & 3 \\ - 1 & - 1 - λ \end{array} | + 0$

Step 5.4

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

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Step 5.4.1

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

$p (λ) = - 4 (- λ + 1) + (1 - λ) ((- 1 - λ) (- 1 - λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2

Simplify the determinant.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

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

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Step 5.4.2.1.1.1

Apply the distributive property.

$p (λ) = - 4 (- λ + 1) + (1 - λ) (- 1 (- 1 - λ) - λ (- 1 - λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.1.2

Apply the distributive property.

$p (λ) = - 4 (- λ + 1) + (1 - λ) (- 1 \cdot - 1 - 1 (- λ) - λ (- 1 - λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.1.3

Apply the distributive property.

$p (λ) = - 4 (- λ + 1) + (1 - λ) (- 1 \cdot - 1 - 1 (- λ) - λ \cdot - 1 - λ (- λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2

Simplify and combine like terms.

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Step 5.4.2.1.2.1

Simplify each term.

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Step 5.4.2.1.2.1.1

Multiply $- 1$ by $- 1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 - 1 (- λ) - λ \cdot - 1 - λ (- λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.2

Multiply $- 1 (- λ)$ .

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Step 5.4.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + 1 λ - λ \cdot - 1 - λ (- λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.2.2

Multiply $λ$ by $1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ - λ \cdot - 1 - λ (- λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.3

Multiply $- λ \cdot - 1$ .

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Step 5.4.2.1.2.1.3.1

Multiply $- 1$ by $- 1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + 1 λ - λ (- λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.3.2

Multiply $λ$ by $1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + λ - λ (- λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + λ - 1 \cdot - 1 λ \cdot λ - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.5

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

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Step 5.4.2.1.2.1.5.1

Move $λ$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + λ - 1 \cdot - 1 (λ \cdot λ) - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + λ - 1 \cdot - 1 λ^{2} - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + λ + 1 λ^{2} - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + λ + λ + λ^{2} - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.2.2

Add $λ$ and $λ$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + 2 λ + λ^{2} - (- 1 \cdot 3)) + 0$

Step 5.4.2.1.3

Multiply $- (- 1 \cdot 3)$ .

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Step 5.4.2.1.3.1

Multiply $- 1$ by $3$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + 2 λ + λ^{2} - - 3) + 0$

Step 5.4.2.1.3.2

Multiply $- 1$ by $- 3$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (1 + 2 λ + λ^{2} + 3) + 0$

Step 5.4.2.2

Add $1$ and $3$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (2 λ + λ^{2} + 4) + 0$

Step 5.4.2.3

Reorder $2 λ$ and $λ^{2}$ .

$p (λ) = - 4 (- λ + 1) + (1 - λ) (λ^{2} + 2 λ + 4) + 0$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Add $- 4 (- λ + 1) + (1 - λ) (λ^{2} + 2 λ + 4)$ and $0$ .

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

Step 5.5.2

Simplify each term.

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Step 5.5.2.1

Apply the distributive property.

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

Step 5.5.2.2

Multiply $- 1$ by $- 4$ .

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

Step 5.5.2.3

Multiply $- 4$ by $1$ .

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

Step 5.5.2.4

Expand $(1 - λ) (λ^{2} + 2 λ + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.5.2.5

Simplify each term.

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Step 5.5.2.5.1

Multiply $λ^{2}$ by $1$ .

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

Step 5.5.2.5.2

Multiply $2 λ$ by $1$ .

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

Step 5.5.2.5.3

Multiply $4$ by $1$ .

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

Step 5.5.2.5.4

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

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Step 5.5.2.5.4.1

Move $λ^{2}$ .

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

Step 5.5.2.5.4.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.5.2.5.4.2.1

Raise $λ$ to the power of $1$ .

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

Step 5.5.2.5.4.2.2

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

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

Step 5.5.2.5.4.3

Add $2$ and $1$ .

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

Step 5.5.2.5.5

Rewrite using the commutative property of multiplication.

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

Step 5.5.2.5.6

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

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Step 5.5.2.5.6.1

Move $λ$ .

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

Step 5.5.2.5.6.2

Multiply $λ$ by $λ$ .

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

Step 5.5.2.5.7

Multiply $- 1$ by $2$ .

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

Step 5.5.2.5.8

Multiply $4$ by $- 1$ .

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

Step 5.5.2.6

Subtract $2 λ^{2}$ from $λ^{2}$ .

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

Step 5.5.2.7

Subtract $4 λ$ from $2 λ$ .

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

Step 5.5.3

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

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Step 5.5.3.1

Add $- 4$ and $4$ .

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

Step 5.5.3.2

Add $4 λ - λ^{2} - 2 λ$ and $0$ .

$p (λ) = 4 λ - λ^{2} - 2 λ - λ^{3}$

Step 5.5.4

Subtract $2 λ$ from $4 λ$ .

$p (λ) = - λ^{2} + 2 λ - λ^{3}$

Step 5.5.5

Move $2 λ$ .

$p (λ) = - λ^{2} - λ^{3} + 2 λ$

Step 5.5.6

Reorder $- λ^{2}$ and $- λ^{3}$ .

$p (λ) = - λ^{3} - λ^{2} + 2 λ$

Step 6

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

$- λ^{3} - λ^{2} + 2 λ = 0$

Step 7

Solve for $λ$ .

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Step 7.1

Factor the left side of the equation.

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Step 7.1.1

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

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Step 7.1.1.1

Factor $- λ$ out of $- λ^{3}$ .

$- λ \cdot λ^{2} - λ^{2} + 2 λ = 0$

Step 7.1.1.2

Factor $- λ$ out of $- λ^{2}$ .

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

Step 7.1.1.3

Factor $- λ$ out of $2 λ$ .

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

Step 7.1.1.4

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

$- λ (λ^{2} + λ) - λ \cdot - 2 = 0$

Step 7.1.1.5

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

$- λ (λ^{2} + λ - 2) = 0$

Step 7.1.2

Factor.

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Step 7.1.2.1

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

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Step 7.1.2.1.1

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

$- 1, 2$

Step 7.1.2.1.2

Write the factored form using these integers.

$- λ ((λ - 1) (λ + 2)) = 0$

Step 7.1.2.2

Remove unnecessary parentheses.

$- λ (λ - 1) (λ + 2) = 0$

Step 7.2

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

$λ = 0$

$λ - 1 = 0$

$λ + 2 = 0$

Step 7.3

Set $λ$ equal to $0$ .

$λ = 0$

Step 7.4

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

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Step 7.4.1

Set $λ - 1$ equal to $0$ .

$λ - 1 = 0$

Step 7.4.2

Add $1$ to both sides of the equation.

$λ = 1$

Step 7.5

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

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Step 7.5.1

Set $λ + 2$ equal to $0$ .

$λ + 2 = 0$

Step 7.5.2

Subtract $2$ from both sides of the equation.

$λ = - 2$

Step 7.6

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

$λ = 0, 1, - 2$

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