Finite Math Examples

$[\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}]$

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 $[\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] - λ I_{3})$

Step 3.2

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

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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.

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply $- λ \cdot 0$ .

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Step 4.1.2.6.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 14 & 45 & 5 \\ 1 & 43 & 6 \\ 2 & 8 & 9 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 + 0 & 5 + 0 \\ 1 + 0 & 43 - λ & 6 + 0 \\ 2 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add $45$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 & 5 + 0 \\ 1 + 0 & 43 - λ & 6 + 0 \\ 2 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 4.3.2

Add $5$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 & 5 \\ 1 + 0 & 43 - λ & 6 + 0 \\ 2 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 4.3.3

Add $1$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 & 5 \\ 1 & 43 - λ & 6 + 0 \\ 2 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 4.3.4

Add $6$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 & 5 \\ 1 & 43 - λ & 6 \\ 2 + 0 & 8 + 0 & 9 - λ \end{array}]$

Step 4.3.5

Add $2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 & 5 \\ 1 & 43 - λ & 6 \\ 2 & 8 + 0 & 9 - λ \end{array}]$

Step 4.3.6

Add $8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 14 - λ & 45 & 5 \\ 1 & 43 - λ & 6 \\ 2 & 8 & 9 - λ \end{array}]$

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 row $1$ 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 $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 43 - λ & 6 \\ 8 & 9 - λ \end{array} |$

Step 5.1.4

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

$(14 - λ) | \begin{array}{c} 43 - λ & 6 \\ 8 & 9 - λ \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} |$

Step 5.1.6

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

$- 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.1.8

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

$5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.1.9

Add the terms together.

$p (λ) = (14 - λ) | \begin{array}{c} 43 - λ & 6 \\ 8 & 9 - λ \end{array} | - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2

Evaluate $| \begin{array}{c} 43 - λ & 6 \\ 8 & 9 - λ \end{array} |$ .

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Step 5.2.1

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

$p (λ) = (14 - λ) ((43 - λ) (9 - λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2

Simplify the determinant.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Expand $(43 - λ) (9 - λ)$ using the FOIL Method.

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Step 5.2.2.1.1.1

Apply the distributive property.

$p (λ) = (14 - λ) (43 (9 - λ) - λ (9 - λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.1.2

Apply the distributive property.

$p (λ) = (14 - λ) (43 \cdot 9 + 43 (- λ) - λ (9 - λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.1.3

Apply the distributive property.

$p (λ) = (14 - λ) (43 \cdot 9 + 43 (- λ) - λ \cdot 9 - λ (- λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2

Simplify and combine like terms.

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Step 5.2.2.1.2.1

Simplify each term.

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Step 5.2.2.1.2.1.1

Multiply $43$ by $9$ .

$p (λ) = (14 - λ) (387 + 43 (- λ) - λ \cdot 9 - λ (- λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.2

Multiply $- 1$ by $43$ .

$p (λ) = (14 - λ) (387 - 43 λ - λ \cdot 9 - λ (- λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.3

Multiply $9$ by $- 1$ .

$p (λ) = (14 - λ) (387 - 43 λ - 9 λ - λ (- λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = (14 - λ) (387 - 43 λ - 9 λ - 1 \cdot - 1 λ \cdot λ - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.5

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

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Step 5.2.2.1.2.1.5.1

Move $λ$ .

$p (λ) = (14 - λ) (387 - 43 λ - 9 λ - 1 \cdot - 1 (λ \cdot λ) - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = (14 - λ) (387 - 43 λ - 9 λ - 1 \cdot - 1 λ^{2} - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = (14 - λ) (387 - 43 λ - 9 λ + 1 λ^{2} - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = (14 - λ) (387 - 43 λ - 9 λ + λ^{2} - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.2.2

Subtract $9 λ$ from $- 43 λ$ .

$p (λ) = (14 - λ) (387 - 52 λ + λ^{2} - 8 \cdot 6) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.1.3

Multiply $- 8$ by $6$ .

$p (λ) = (14 - λ) (387 - 52 λ + λ^{2} - 48) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.2

Subtract $48$ from $387$ .

$p (λ) = (14 - λ) (- 52 λ + λ^{2} + 339) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.2.2.3

Reorder $- 52 λ$ and $λ^{2}$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 | \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} | + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.3

Evaluate $| \begin{array}{c} 1 & 6 \\ 2 & 9 - λ \end{array} |$ .

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Step 5.3.1

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

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (1 (9 - λ) - 2 \cdot 6) + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

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 $9 - λ$ by $1$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (9 - λ - 2 \cdot 6) + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.3.2.1.2

Multiply $- 2$ by $6$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (9 - λ - 12) + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.3.2.2

Subtract $12$ from $9$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 | \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$

Step 5.4

Evaluate $| \begin{array}{c} 1 & 43 - λ \\ 2 & 8 \end{array} |$ .

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Step 5.4.1

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

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (1 \cdot 8 - 2 (43 - λ))$

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

Multiply $8$ by $1$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (8 - 2 (43 - λ))$

Step 5.4.2.1.2

Apply the distributive property.

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (8 - 2 \cdot 43 - 2 (- λ))$

Step 5.4.2.1.3

Multiply $- 2$ by $43$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (8 - 86 - 2 (- λ))$

Step 5.4.2.1.4

Multiply $- 1$ by $- 2$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (8 - 86 + 2 λ)$

Step 5.4.2.2

Subtract $86$ from $8$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (- 78 + 2 λ)$

Step 5.4.2.3

Reorder $- 78$ and $2 λ$ .

$p (λ) = (14 - λ) (λ^{2} - 52 λ + 339) - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Expand $(14 - λ) (λ^{2} - 52 λ + 339)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = 14 λ^{2} + 14 (- 52 λ) + 14 \cdot 339 - λ \cdot λ^{2} - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2

Simplify each term.

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Step 5.5.1.2.1

Multiply $- 52$ by $14$ .

$p (λ) = 14 λ^{2} - 728 λ + 14 \cdot 339 - λ \cdot λ^{2} - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.2

Multiply $14$ by $339$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ \cdot λ^{2} - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.3

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

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Step 5.5.1.2.3.1

Move $λ^{2}$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - (λ^{2} λ) - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.3.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.5.1.2.3.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - (λ^{2} λ^{1}) - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.3.2.2

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

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{2 + 1} - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.3.3

Add $2$ and $1$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{3} - λ (- 52 λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.4

Rewrite using the commutative property of multiplication.

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{3} - 1 \cdot - 52 λ \cdot λ - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.5

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

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Step 5.5.1.2.5.1

Move $λ$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{3} - 1 \cdot - 52 (λ \cdot λ) - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.5.2

Multiply $λ$ by $λ$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{3} - 1 \cdot - 52 λ^{2} - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.6

Multiply $- 1$ by $- 52$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{3} + 52 λ^{2} - λ \cdot 339 - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.2.7

Multiply $339$ by $- 1$ .

$p (λ) = 14 λ^{2} - 728 λ + 4746 - λ^{3} + 52 λ^{2} - 339 λ - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.3

Add $14 λ^{2}$ and $52 λ^{2}$ .

$p (λ) = 66 λ^{2} - 728 λ + 4746 - λ^{3} - 339 λ - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.4

Subtract $339 λ$ from $- 728 λ$ .

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} - 45 (- λ - 3) + 5 (2 λ - 78)$

Step 5.5.1.5

Apply the distributive property.

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} - 45 (- λ) - 45 \cdot - 3 + 5 (2 λ - 78)$

Step 5.5.1.6

Multiply $- 1$ by $- 45$ .

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} + 45 λ - 45 \cdot - 3 + 5 (2 λ - 78)$

Step 5.5.1.7

Multiply $- 45$ by $- 3$ .

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} + 45 λ + 135 + 5 (2 λ - 78)$

Step 5.5.1.8

Apply the distributive property.

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} + 45 λ + 135 + 5 (2 λ) + 5 \cdot - 78$

Step 5.5.1.9

Multiply $2$ by $5$ .

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} + 45 λ + 135 + 10 λ + 5 \cdot - 78$

Step 5.5.1.10

Multiply $5$ by $- 78$ .

$p (λ) = 66 λ^{2} - 1067 λ + 4746 - λ^{3} + 45 λ + 135 + 10 λ - 390$

Step 5.5.2

Add $- 1067 λ$ and $45 λ$ .

$p (λ) = 66 λ^{2} - 1022 λ + 4746 - λ^{3} + 135 + 10 λ - 390$

Step 5.5.3

Add $- 1022 λ$ and $10 λ$ .

$p (λ) = 66 λ^{2} - 1012 λ + 4746 - λ^{3} + 135 - 390$

Step 5.5.4

Add $4746$ and $135$ .

$p (λ) = 66 λ^{2} - 1012 λ - λ^{3} + 4881 - 390$

Step 5.5.5

Subtract $390$ from $4881$ .

$p (λ) = 66 λ^{2} - 1012 λ - λ^{3} + 4491$

Step 5.5.6

Move $- 1012 λ$ .

$p (λ) = 66 λ^{2} - λ^{3} - 1012 λ + 4491$

Step 5.5.7

Reorder $66 λ^{2}$ and $- λ^{3}$ .

$p (λ) = - λ^{3} + 66 λ^{2} - 1012 λ + 4491$

Step 6

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

$- λ^{3} + 66 λ^{2} - 1012 λ + 4491 = 0$

Step 7

Solve for $λ$ .

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Step 7.1

Factor $- λ^{3} + 66 λ^{2} - 1012 λ + 4491$ using the rational roots test.

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Step 7.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 4491, \pm 3, \pm 1497, \pm 9, \pm 499$

$q = \pm 1$

Step 7.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 4491, \pm 3, \pm 1497, \pm 9, \pm 499$

Step 7.1.3

Substitute $9$ and simplify the expression. In this case, the expression is equal to $0$ so $9$ is a root of the polynomial.

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Step 7.1.3.1

Substitute $9$ into the polynomial.

$- 9^{3} + 66 \cdot 9^{2} - 1012 \cdot 9 + 4491$

Step 7.1.3.2

Raise $9$ to the power of $3$ .

$- 1 \cdot 729 + 66 \cdot 9^{2} - 1012 \cdot 9 + 4491$

Step 7.1.3.3

Multiply $- 1$ by $729$ .

$- 729 + 66 \cdot 9^{2} - 1012 \cdot 9 + 4491$

Step 7.1.3.4

Raise $9$ to the power of $2$ .

$- 729 + 66 \cdot 81 - 1012 \cdot 9 + 4491$

Step 7.1.3.5

Multiply $66$ by $81$ .

$- 729 + 5346 - 1012 \cdot 9 + 4491$

Step 7.1.3.6

Add $- 729$ and $5346$ .

$4617 - 1012 \cdot 9 + 4491$

Step 7.1.3.7

Multiply $- 1012$ by $9$ .

$4617 - 9108 + 4491$

Step 7.1.3.8

Subtract $9108$ from $4617$ .

$- 4491 + 4491$

Step 7.1.3.9

Add $- 4491$ and $4491$ .

$0$

Step 7.1.4

Since $9$ is a known root, divide the polynomial by $λ - 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- λ^{3} + 66 λ^{2} - 1012 λ + 4491}{λ - 9}$

Step 7.1.5

Divide $- λ^{3} + 66 λ^{2} - 1012 λ + 4491$ by $λ - 9$ .

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Step 7.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$λ$

$9$

$λ^{3}$

$66 λ^{2}$

$1012 λ$

$4491$

Step 7.1.5.2

Divide the highest order term in the dividend $- λ^{3}$ by the highest order term in divisor $λ$ .

				-	$λ^{2}$
	$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$

Step 7.1.5.3

Multiply the new quotient term by the divisor.

			-	$λ^{2}$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			-	$λ^{3}$	+	$9 λ^{2}$

Step 7.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- λ^{3} + 9 λ^{2}$

			-	$λ^{2}$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$

Step 7.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$λ^{2}$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$

Step 7.1.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$λ^{2}$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$

Step 7.1.5.7

Divide the highest order term in the dividend $57 λ^{2}$ by the highest order term in divisor $λ$ .

			-	$λ^{2}$	+	$57 λ$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$

Step 7.1.5.8

Multiply the new quotient term by the divisor.

			-	$λ^{2}$	+	$57 λ$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					+	$57 λ^{2}$	-	$513 λ$

Step 7.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $57 λ^{2} - 513 λ$

			-	$λ^{2}$	+	$57 λ$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$

Step 7.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$λ^{2}$	+	$57 λ$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$
							-	$499 λ$

Step 7.1.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$λ^{2}$	+	$57 λ$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$
							-	$499 λ$	+	$4491$

Step 7.1.5.12

Divide the highest order term in the dividend $- 499 λ$ by the highest order term in divisor $λ$ .

			-	$λ^{2}$	+	$57 λ$	-	$499$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$
							-	$499 λ$	+	$4491$

Step 7.1.5.13

Multiply the new quotient term by the divisor.

			-	$λ^{2}$	+	$57 λ$	-	$499$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$
							-	$499 λ$	+	$4491$
							-	$499 λ$	+	$4491$

Step 7.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 499 λ + 4491$

			-	$λ^{2}$	+	$57 λ$	-	$499$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$
							-	$499 λ$	+	$4491$
							+	$499 λ$	-	$4491$

Step 7.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$λ^{2}$	+	$57 λ$	-	$499$
$λ$	-	$9$	-	$λ^{3}$	+	$66 λ^{2}$	-	$1012 λ$	+	$4491$
			+	$λ^{3}$	-	$9 λ^{2}$
					+	$57 λ^{2}$	-	$1012 λ$
					-	$57 λ^{2}$	+	$513 λ$
							-	$499 λ$	+	$4491$
							+	$499 λ$	-	$4491$
										$0$

Step 7.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$- λ^{2} + 57 λ - 499$

Step 7.1.6

Write $- λ^{3} + 66 λ^{2} - 1012 λ + 4491$ as a set of factors.

$(λ - 9) (- λ^{2} + 57 λ - 499) = 0$

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 $0$ .

$λ - 9 = 0$

$- λ^{2} + 57 λ - 499 = 0$

Step 7.3

Set $λ - 9$ equal to $0$ and solve for $λ$ .

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Step 7.3.1

Set $λ - 9$ equal to $0$ .

$λ - 9 = 0$

Step 7.3.2

Add $9$ to both sides of the equation.

$λ = 9$

Step 7.4

Set $- λ^{2} + 57 λ - 499$ equal to $0$ and solve for $λ$ .

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Step 7.4.1

Set $- λ^{2} + 57 λ - 499$ equal to $0$ .

$- λ^{2} + 57 λ - 499 = 0$

Step 7.4.2

Solve $- λ^{2} + 57 λ - 499 = 0$ for $λ$ .

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Step 7.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.4.2.2

Substitute the values $a = - 1$ , $b = 57$ , and $c = - 499$ into the quadratic formula and solve for $λ$ .

$\frac{- 57 \pm \sqrt{57^{2} - 4 \cdot (- 1 \cdot - 499)}}{2 \cdot - 1}$

Step 7.4.2.3

Simplify.

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Step 7.4.2.3.1

Simplify the numerator.

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Step 7.4.2.3.1.1

Raise $57$ to the power of $2$ .

$λ = \frac{- 57 \pm \sqrt{3249 - 4 \cdot - 1 \cdot - 499}}{2 \cdot - 1}$

Step 7.4.2.3.1.2

Multiply $- 4 \cdot - 1 \cdot - 499$ .

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Step 7.4.2.3.1.2.1

Multiply $- 4$ by $- 1$ .

$λ = \frac{- 57 \pm \sqrt{3249 + 4 \cdot - 499}}{2 \cdot - 1}$

Step 7.4.2.3.1.2.2

Multiply $4$ by $- 499$ .

$λ = \frac{- 57 \pm \sqrt{3249 - 1996}}{2 \cdot - 1}$

Step 7.4.2.3.1.3

Subtract $1996$ from $3249$ .

$λ = \frac{- 57 \pm \sqrt{1253}}{2 \cdot - 1}$

Step 7.4.2.3.2

Multiply $2$ by $- 1$ .

$λ = \frac{- 57 \pm \sqrt{1253}}{- 2}$

Step 7.4.2.3.3

Simplify $\frac{- 57 \pm \sqrt{1253}}{- 2}$ .

$λ = \frac{57 \pm \sqrt{1253}}{2}$

Step 7.4.2.4

The final answer is the combination of both solutions.

$λ = \frac{57 + \sqrt{1253}}{2}, \frac{57 - \sqrt{1253}}{2}$

Step 7.5

The final solution is all the values that make $(λ - 9) (- λ^{2} + 57 λ - 499) = 0$ true.

$λ = 9, \frac{57 + \sqrt{1253}}{2}, \frac{57 - \sqrt{1253}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$λ = 9, \frac{57 + \sqrt{1253}}{2}, \frac{57 - \sqrt{1253}}{2}$

Decimal Form:

$λ = 9, 46.19887002 \dots, 10.80112997 \dots$