Linear Algebra Examples

$[\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}]$

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

$p (λ) = determinant (A + (- λ I_{3}))$

Substitute the known values in the formula.

$p (λ) = determinant ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Subtract the eigenvalue times the identity matrix from the original matrix.

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

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify each element of the matrix $[\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$ .

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Simplify element $0, 0$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $0, 1$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $0, 2$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 0$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 1$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $1, 2$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 0$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 1$ by multiplying $- λ \cdot 0$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Simplify element $2, 2$ by multiplying $- λ \cdot 1$ .

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Combine the similar matrices with each others.

$p (λ) = ([\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Simplify each element of the matrix $[\begin{array}{c} - λ + 2 & 0 + 2 & 0 + 1 \\ 0 + 1 & - λ - 9 & 0 + 0 \\ 0 + 1 & 0 + 2 & - λ + 1 \end{array}]$ .

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Combine the same size matrices $[\begin{array}{c} 2 & 2 & 1 \\ 1 & - 9 & 0 \\ 1 & 2 & 1 \end{array}]$ and $[\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}]$ by adding the corresponding elements of each.

$p (λ) = ([\begin{array}{c} - λ + 2 & 0 + 2 & 0 + 1 \\ 0 + 1 & - λ - 9 & 0 + 0 \\ 0 + 1 & 0 + 2 & - λ + 1 \end{array}])$

Simplify element $0, 1$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ + 2 & 2 & 0 + 1 \\ 0 + 1 & - λ - 9 & 0 + 0 \\ 0 + 1 & 0 + 2 & - λ + 1 \end{array}])$

Simplify element $0, 2$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ + 2 & 2 & 1 \\ 0 + 1 & - λ - 9 & 0 + 0 \\ 0 + 1 & 0 + 2 & - λ + 1 \end{array}])$

Simplify element $1, 0$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ + 2 & 2 & 1 \\ 1 & - λ - 9 & 0 + 0 \\ 0 + 1 & 0 + 2 & - λ + 1 \end{array}])$

Simplify element $1, 2$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ + 2 & 2 & 1 \\ 1 & - λ - 9 & 0 \\ 0 + 1 & 0 + 2 & - λ + 1 \end{array}])$

Simplify element $2, 0$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ + 2 & 2 & 1 \\ 1 & - λ - 9 & 0 \\ 1 & 0 + 2 & - λ + 1 \end{array}])$

Simplify element $2, 1$ of the matrix.

$p (λ) = ([\begin{array}{c} - λ + 2 & 2 & 1 \\ 1 & - λ - 9 & 0 \\ 1 & 2 & - λ + 1 \end{array}])$

The determinant of $[\begin{array}{c} - λ + 2 & 2 & 1 \\ 1 & - λ - 9 & 0 \\ 1 & 2 & - λ + 1 \end{array}]$ is $- λ^{3} - 6 λ^{2} + 28 λ - 9$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = 1 | \begin{array}{c} 1 & - λ - 9 \\ 1 & 2 \end{array} | + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

The determinant of $1 | \begin{array}{c} 1 & - λ - 9 \\ 1 & 2 \end{array} |$ is $λ + 11$ .

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

$p (λ) = (1) (2) - 1 (- λ - 9) + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $2$ by $1$ .

$p (λ) = 2 - 1 (- λ - 9) + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Apply the distributive property.

$p (λ) = 2 + (- 1 (- λ) - 1 \cdot - 9) + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Simplify $- 1 (- λ)$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = 2 + (1 λ - 1 \cdot - 9) + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Multiply $λ$ by $1$ .

$p (λ) = 2 + (λ - 1 \cdot - 9) + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Multiply $- 1$ by $- 9$ .

$p (λ) = 2 + (λ + 9) + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Simplify the expression.

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Remove unnecessary parentheses.

$p (λ) = 2 + λ + 9 + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Add $2$ and $9$ .

$p (λ) = λ + 11 + 0 | \begin{array}{c} - λ + 2 & 2 \\ 1 & 2 \end{array} | + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

Since the matrix is multiplied by $0$ , the determinant is $0$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$

The determinant of $(- λ + 1) | \begin{array}{c} - λ + 2 & 2 \\ 1 & - λ - 9 \end{array} |$ is $- λ^{3} - 6 λ^{2} + 27 λ - 20$ .

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

$p (λ) = λ + 11 + 0 (- λ + 1) ((- λ + 2) (- λ - 9) - 1 \cdot 2)$

Simplify the determinant.

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Simplify terms.

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Simplify each term.

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Expand $(- λ + 2) (- λ - 9)$ using the FOIL Method.

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Apply the distributive property.

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ (- λ - 9) + 2 (- λ - 9) - 1 \cdot 2)$

Apply the distributive property.

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ (- λ) - λ \cdot - 9 + 2 (- λ - 9) - 1 \cdot 2)$

Apply the distributive property.

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ (- λ) - λ \cdot - 9 + (2 (- λ) + 2 \cdot - 9) - 1 \cdot 2)$

Remove parentheses.

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ (- λ) - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Simplify and combine like terms.

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Simplify each term.

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

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Move $λ$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ λ \cdot - 1 - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ λ \cdot - 1 - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Raise $λ$ to the power of $1$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ λ \cdot - 1 - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

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

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ^{1 + 1} \cdot - 1 - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Add $1$ and $1$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (- λ^{2} \cdot - 1 - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Simplify $- λ^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (1 λ^{2} - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Multiply $λ^{2}$ by $1$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} - λ \cdot - 9 + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Multiply $- 9$ by $- 1$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} + 9 λ + 2 (- λ) + 2 \cdot - 9 - 1 \cdot 2)$

Multiply $- 1$ by $2$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} + 9 λ - 2 λ + 2 \cdot - 9 - 1 \cdot 2)$

Multiply $2$ by $- 9$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} + 9 λ - 2 λ - 18 - 1 \cdot 2)$

Subtract $2 λ$ from $9 λ$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} + 7 λ - 18 - 1 \cdot 2)$

Multiply $- 1$ by $2$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} + 7 λ - 18 - 2)$

Subtract $2$ from $- 18$ .

$p (λ) = λ + 11 + 0 + (- λ + 1) (λ^{2} + 7 λ - 20)$

Expand $(- λ + 1) (λ^{2} + 7 λ - 20)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = λ + 11 + 0 - λ λ^{2} - λ (7 λ) - λ \cdot - 20 + (1 λ^{2} + 1 (7 λ) + 1 \cdot - 20)$

Simplify terms.

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Remove unnecessary parentheses.

$p (λ) = λ + 11 + 0 - λ λ^{2} - λ (7 λ) - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Simplify each term.

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

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Move $λ^{2}$ .

$p (λ) = λ + 11 + 0 - (λ^{2} λ) - λ (7 λ) - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Combine $λ^{2}$ and $λ$

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Raise $λ$ to the power of $1$ .

$p (λ) = λ + 11 + 0 - (λ^{2} λ) - λ (7 λ) - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

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

$p (λ) = λ + 11 + 0 - λ^{2 + 1} - λ (7 λ) - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Add $2$ and $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - λ (7 λ) - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

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

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Move $λ$ .

$p (λ) = λ + 11 + 0 - λ^{3} - λ λ \cdot 7 - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Combine $λ$ and $λ$

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Raise $λ$ to the power of $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - λ λ \cdot 7 - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Raise $λ$ to the power of $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - λ λ \cdot 7 - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

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

$p (λ) = λ + 11 + 0 - λ^{3} - λ^{1 + 1} \cdot 7 - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Add $1$ and $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - λ^{2} \cdot 7 - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Multiply $7$ by $- 1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 7 λ^{2} - λ \cdot - 20 + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Multiply $- 20$ by $- 1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 7 λ^{2} + 20 λ + 1 λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Multiply $λ^{2}$ by $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 7 λ^{2} + 20 λ + λ^{2} + 1 (7 λ) + 1 \cdot - 20$

Multiply $7 λ$ by $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 7 λ^{2} + 20 λ + λ^{2} + 7 λ + 1 \cdot - 20$

Multiply $- 20$ by $1$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 7 λ^{2} + 20 λ + λ^{2} + 7 λ - 20$

Simplify by adding terms.

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Add $- 7 λ^{2}$ and $λ^{2}$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 6 λ^{2} + 20 λ + 7 λ - 20$

Add $20 λ$ and $7 λ$ .

$p (λ) = λ + 11 + 0 - λ^{3} - 6 λ^{2} + 27 λ - 20$

Add $λ$ and $0$ .

$p (λ) = λ + 11 - λ^{3} - 6 λ^{2} + 27 λ - 20$

Add $λ$ and $27 λ$ .

$p (λ) = 11 - λ^{3} - 6 λ^{2} + 28 λ - 20$

Subtract $20$ from $11$ .

$p (λ) = - λ^{3} - 6 λ^{2} + 28 λ - 9$

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

$- λ^{3} - 6 λ^{2} + 28 λ - 9 = 0$

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

$λ \approx - 9.16297121, 0.34905606, 2.81391514$

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