Algebra Examples

$[\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}]$

Step 1

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

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

Step 2

Substitute the known values in the formula.

$p (λ) = determinant ([\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 3

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

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

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Simplify each element in the matrix.

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Rearrange $- λ \cdot 1$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 0$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 0$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 0$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 1$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 0$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 0$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 0$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}]$

Rearrange $- λ \cdot 1$ .

$p (λ) = [\begin{array}{c} 9 & 8 & 7 \\ 3 & 4 & 5 \\ 2 & 1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}]$

Add the corresponding elements.

$p (λ) = [\begin{array}{c} 9 - λ & 8 + 0 & 7 + 0 \\ 3 + 0 & 4 - λ & 5 + 0 \\ 2 + 0 & 1 + 0 & 0 - λ \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 9 - λ & 8 + 0 & 7 + 0 \\ 3 + 0 & 4 - λ & 5 + 0 \\ 2 + 0 & 1 + 0 & 0 - λ \end{array}]$ .

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Simplify $8 + 0$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 + 0 \\ 3 + 0 & 4 - λ & 5 + 0 \\ 2 + 0 & 1 + 0 & 0 - λ \end{array}]$

Simplify $7 + 0$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 \\ 3 + 0 & 4 - λ & 5 + 0 \\ 2 + 0 & 1 + 0 & 0 - λ \end{array}]$

Simplify $3 + 0$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 \\ 3 & 4 - λ & 5 + 0 \\ 2 + 0 & 1 + 0 & 0 - λ \end{array}]$

Simplify $5 + 0$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 \\ 3 & 4 - λ & 5 \\ 2 + 0 & 1 + 0 & 0 - λ \end{array}]$

Simplify $2 + 0$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 \\ 3 & 4 - λ & 5 \\ 2 & 1 + 0 & 0 - λ \end{array}]$

Simplify $1 + 0$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 \\ 3 & 4 - λ & 5 \\ 2 & 1 & 0 - λ \end{array}]$

Simplify $0 - λ$ .

$p (λ) = [\begin{array}{c} 9 - λ & 8 & 7 \\ 3 & 4 - λ & 5 \\ 2 & 1 & - λ \end{array}]$

Step 4

Find the determinant of $[\begin{array}{c} 9 - λ & 8 & 7 \\ 3 & 4 - λ & 5 \\ 2 & 1 & - λ \end{array}]$ .

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Set up the determinant by breaking it into smaller components.

$p (λ) = (9 - λ) | \begin{array}{c} 4 - λ & 5 \\ 1 & - λ \end{array} | - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Find the determinant of $(9 - λ) | \begin{array}{c} 4 - λ & 5 \\ 1 & - λ \end{array} |$ .

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

$p (λ) = (9 - λ) ((4 - λ) (- λ) - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Apply the distributive property.

$p (λ) = (9 - λ) (4 (- λ) - λ (- λ) - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 1$ by $4$ .

$p (λ) = (9 - λ) (- 4 λ - λ (- λ) - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Rewrite using the commutative property of multiplication.

$p (λ) = (9 - λ) (- 4 λ - 1 \cdot (- 1 λ \cdot λ) - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Simplify each term.

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

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Move $λ$ .

$p (λ) = (9 - λ) (- 4 λ - 1 \cdot (- 1 (λ \cdot λ)) - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $λ$ by $λ$ .

$p (λ) = (9 - λ) (- 4 λ - 1 \cdot (- 1 λ^{2}) - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 1$ by $- 1$ .

$p (λ) = (9 - λ) (- 4 λ + 1 λ^{2} - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $λ^{2}$ by $1$ .

$p (λ) = (9 - λ) (- 4 λ + λ^{2} - 1 \cdot 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 1$ by $5$ .

$p (λ) = (9 - λ) (- 4 λ + λ^{2} - 5) - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Expand $(9 - λ) (- 4 λ + λ^{2} - 5)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = 9 (- 4 λ) + 9 λ^{2} + 9 \cdot - 5 - λ (- 4 λ) - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Simplify terms.

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Simplify each term.

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Multiply $- 4$ by $9$ .

$p (λ) = - 36 λ + 9 λ^{2} + 9 \cdot - 5 - λ (- 4 λ) - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $9$ by $- 5$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 - λ (- 4 λ) - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Rewrite using the commutative property of multiplication.

$p (λ) = - 36 λ + 9 λ^{2} - 45 - 1 \cdot (- 4 λ \cdot λ) - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

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

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Move $λ$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 - 1 \cdot (- 4 (λ \cdot λ)) - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $λ$ by $λ$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 - 1 \cdot (- 4 λ^{2}) - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 1$ by $- 4$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 + 4 λ^{2} - λ \cdot λ^{2} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

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

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Move $λ^{2}$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 + 4 λ^{2} - (λ^{2} λ) - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $λ^{2}$ by $λ$ .

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Raise $λ$ to the power of $1$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 + 4 λ^{2} - (λ^{2} λ) - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

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

$p (λ) = - 36 λ + 9 λ^{2} - 45 + 4 λ^{2} - λ^{2 + 1} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Add $2$ and $1$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 + 4 λ^{2} - λ^{3} - λ \cdot - 5 - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 5$ by $- 1$ .

$p (λ) = - 36 λ + 9 λ^{2} - 45 + 4 λ^{2} - λ^{3} + 5 λ - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Simplify by adding terms.

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Add $- 36 λ$ and $5 λ$ .

$p (λ) = 9 λ^{2} - 45 + 4 λ^{2} - λ^{3} - 31 λ - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Add $9 λ^{2}$ and $4 λ^{2}$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ - 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} | + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Find the determinant of $- 3 | \begin{array}{c} 8 & 7 \\ 1 & - λ \end{array} |$ .

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

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ - 3 ((8) (- λ) - 1 \cdot 7) + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $- 1$ by $8$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ - 3 (- 8 λ - 1 \cdot 7) + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 1$ by $7$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ - 3 (- 8 λ - 7) + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Simplify by multiplying through.

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Apply the distributive property.

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ - 3 (- 8 λ) - 3 \cdot - 7 + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply.

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Multiply $- 8$ by $- 3$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ - 3 \cdot - 7 + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Multiply $- 3$ by $- 7$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$

Find the determinant of $2 | \begin{array}{c} 8 & 7 \\ 4 - λ & 5 \end{array} |$ .

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

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 ((8) (5) - (4 - λ) \cdot 7)$

Simplify the determinant.

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Simplify each term.

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Multiply $8$ by $5$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 - (4 - λ) \cdot 7)$

Apply the distributive property.

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 + (- 1 \cdot 4 + λ) \cdot 7)$

Multiply $- 1$ by $4$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 + (- 4 + λ) \cdot 7)$

Multiply $- - λ$ .

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Multiply $- 1$ by $- 1$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 + (- 4 + 1 λ) \cdot 7)$

Multiply $λ$ by $1$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 + (- 4 + λ) \cdot 7)$

Apply the distributive property.

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 - 4 \cdot 7 + λ \cdot 7)$

Multiply $- 4$ by $7$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 - 28 + λ \cdot 7)$

Move $7$ to the left of $λ$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (40 - 28 + 7 λ)$

Simplify by multiplying through.

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Subtract $28$ from $40$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 (12 + 7 λ)$

Apply the distributive property.

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 2 \cdot 12 + 2 (7 λ)$

Multiply.

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Multiply $2$ by $12$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 24 + 2 (7 λ)$

Multiply $7$ by $2$ .

$p (λ) = 13 λ^{2} - 45 - λ^{3} - 31 λ + 24 λ + 21 + 24 + 14 λ$

Add $- 45$ and $21$ .

$p (λ) = 13 λ^{2} - λ^{3} - 31 λ + 24 λ - 24 + 24 + 14 λ$

Combine the opposite terms in $13 λ^{2} - λ^{3} - 31 λ + 24 λ - 24 + 24 + 14 λ$ .

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Add $- 24$ and $24$ .

$p (λ) = 13 λ^{2} - λ^{3} - 31 λ + 24 λ + 0 + 14 λ$

Add $13 λ^{2} - λ^{3} - 31 λ + 24 λ$ and $0$ .

$p (λ) = 13 λ^{2} - λ^{3} - 31 λ + 24 λ + 14 λ$

Add $- 31 λ$ and $24 λ$ .

$p (λ) = 13 λ^{2} - λ^{3} - 7 λ + 14 λ$

Add $- 7 λ$ and $14 λ$ .

$p (λ) = 13 λ^{2} - λ^{3} + 7 λ$

Factor the characteristic polynomial.

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Factor $λ$ out of $13 λ^{2} - λ^{3} + 7 λ$ .

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Factor $λ$ out of $13 λ^{2}$ .

$p (λ) = λ (13 λ) - λ^{3} + 7 λ$

Factor $λ$ out of $- λ^{3}$ .

$p (λ) = λ (13 λ) + λ (- λ^{2}) + 7 λ$

Factor $λ$ out of $7 λ$ .

$p (λ) = λ (13 λ) + λ (- λ^{2}) + λ \cdot 7$

Factor $λ$ out of $λ (13 λ) + λ (- λ^{2})$ .

$p (λ) = λ (13 λ - λ^{2}) + λ \cdot 7$

Factor $λ$ out of $λ (13 λ - λ^{2}) + λ \cdot 7$ .

$p (λ) = λ (13 λ - λ^{2} + 7)$

Reorder terms.

$p (λ) = λ (- λ^{2} + 13 λ + 7)$

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