Linear Algebra Examples

$[\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}]$

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 $[\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}]$ for $A$ .

Step 3.2

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

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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.

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply $- λ \cdot 0$ .

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Step 4.1.2.6.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} \frac{31}{6} & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} & 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} \frac{31}{6} - λ & - \frac{73}{18} + 0 & - \frac{16}{9} + 0 \\ - \frac{73}{18} + 0 & \frac{91}{18} - λ & \frac{16}{9} + 0 \\ - \frac{16}{9} + 0 & \frac{16}{9} + 0 & 2 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add $- \frac{73}{18}$ and $0$ .

$p (λ) = determinant [\begin{array}{c} \frac{31}{6} - λ & - \frac{73}{18} & - \frac{16}{9} + 0 \\ - \frac{73}{18} + 0 & \frac{91}{18} - λ & \frac{16}{9} + 0 \\ - \frac{16}{9} + 0 & \frac{16}{9} + 0 & 2 - λ \end{array}]$

Step 4.3.2

Add $- \frac{16}{9}$ and $0$ .

$p (λ) = determinant [\begin{array}{c} \frac{31}{6} - λ & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} + 0 & \frac{91}{18} - λ & \frac{16}{9} + 0 \\ - \frac{16}{9} + 0 & \frac{16}{9} + 0 & 2 - λ \end{array}]$

Step 4.3.3

Add $- \frac{73}{18}$ and $0$ .

$p (λ) = determinant [\begin{array}{c} \frac{31}{6} - λ & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} - λ & \frac{16}{9} + 0 \\ - \frac{16}{9} + 0 & \frac{16}{9} + 0 & 2 - λ \end{array}]$

Step 4.3.4

Add $\frac{16}{9}$ and $0$ .

$p (λ) = determinant [\begin{array}{c} \frac{31}{6} - λ & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} - λ & \frac{16}{9} \\ - \frac{16}{9} + 0 & \frac{16}{9} + 0 & 2 - λ \end{array}]$

Step 4.3.5

Add $- \frac{16}{9}$ and $0$ .

$p (λ) = determinant [\begin{array}{c} \frac{31}{6} - λ & - \frac{73}{18} & - \frac{16}{9} \\ - \frac{73}{18} & \frac{91}{18} - λ & \frac{16}{9} \\ - \frac{16}{9} & \frac{16}{9} + 0 & 2 - λ \end{array}]$

Step 4.3.6

Add $\frac{16}{9}$ and $0$ .

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.

$| \begin{array}{c} \frac{91}{18} - λ & \frac{16}{9} \\ \frac{16}{9} & 2 - λ \end{array} |$

Step 5.1.4

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

$(\frac{31}{6} - λ) | \begin{array}{c} \frac{91}{18} - λ & \frac{16}{9} \\ \frac{16}{9} & 2 - λ \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} |$

Step 5.1.6

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

$\frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.1.8

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

$- \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.1.9

Add the terms together.

$p (λ) = (\frac{31}{6} - λ) | \begin{array}{c} \frac{91}{18} - λ & \frac{16}{9} \\ \frac{16}{9} & 2 - λ \end{array} | + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2

Evaluate $| \begin{array}{c} \frac{91}{18} - λ & \frac{16}{9} \\ \frac{16}{9} & 2 - λ \end{array} |$ .

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Step 5.2.1

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

$p (λ) = (\frac{31}{6} - λ) ((\frac{91}{18} - λ) (2 - λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

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 $(\frac{91}{18} - λ) (2 - λ)$ using the FOIL Method.

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Step 5.2.2.1.1.1

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{18} (2 - λ) - λ (2 - λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.1.2

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{18} \cdot 2 + \frac{91}{18} (- λ) - λ (2 - λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.1.3

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{18} \cdot 2 + \frac{91}{18} (- λ) - λ \cdot 2 - λ (- λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

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

Cancel the common factor of $2$ .

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Step 5.2.2.1.2.1.1.1

Factor $2$ out of $18$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{2 (9)} \cdot 2 + \frac{91}{18} (- λ) - λ \cdot 2 - λ (- λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.1.2

Cancel the common factor.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{2 \cdot 9} \cdot 2 + \frac{91}{18} (- λ) - λ \cdot 2 - λ (- λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.1.3

Rewrite the expression.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} + \frac{91}{18} (- λ) - λ \cdot 2 - λ (- λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.2

Combine $\frac{91}{18}$ and $λ$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - λ \cdot 2 - λ (- λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.3

Multiply $2$ by $- 1$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ - λ (- λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ - 1 \cdot - 1 λ \cdot λ - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

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 $λ$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ - 1 \cdot - 1 (λ \cdot λ) - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ - 1 \cdot - 1 λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ + 1 λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.2

To write $- 2 λ$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} - 2 λ \cdot \frac{18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.3

Combine $- 2 λ$ and $\frac{18}{18}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} - \frac{91 λ}{18} + \frac{- 2 λ \cdot 18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.4

Combine the numerators over the common denominator.

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} + \frac{- 91 λ - 2 λ \cdot 18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.5

To write $\frac{91}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91}{9} \cdot \frac{2}{2} + \frac{- 91 λ - 2 λ \cdot 18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.6

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.1.2.6.1

Multiply $\frac{91}{9}$ by $\frac{2}{2}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91 \cdot 2}{9 \cdot 2} + \frac{- 91 λ - 2 λ \cdot 18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.6.2

Multiply $9$ by $2$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91 \cdot 2}{18} + \frac{- 91 λ - 2 λ \cdot 18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.7

Combine the numerators over the common denominator.

$p (λ) = (\frac{31}{6} - λ) (\frac{91 \cdot 2 - 91 λ - 2 λ \cdot 18}{18} + λ^{2} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.8

To write $λ^{2}$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91 \cdot 2 - 91 λ - 2 λ \cdot 18}{18} + λ^{2} \cdot \frac{18}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.9

Combine $λ^{2}$ and $\frac{18}{18}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{91 \cdot 2 - 91 λ - 2 λ \cdot 18}{18} + \frac{λ^{2} \cdot 18}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.2.10

Combine the numerators over the common denominator.

$p (λ) = (\frac{31}{6} - λ) (\frac{91 \cdot 2 - 91 λ - 2 λ \cdot 18 + λ^{2} \cdot 18}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3

Simplify the numerator.

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Step 5.2.2.1.3.1

Multiply $91$ by $2$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{182 - 91 λ - 2 λ \cdot 18 + λ^{2} \cdot 18}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.2

Multiply $18$ by $- 2$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{182 - 91 λ - 36 λ + λ^{2} \cdot 18}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.3

Move $18$ to the left of $λ^{2}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{182 - 91 λ - 36 λ + 18 \cdot λ^{2}}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.4

Subtract $36 λ$ from $- 91 λ$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{182 - 127 λ + 18 λ^{2}}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5

Factor by grouping.

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Step 5.2.2.1.3.5.1

Reorder terms.

$p (λ) = (\frac{31}{6} - λ) (\frac{18 λ^{2} - 127 λ + 182}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 18 \cdot 182 = 3276$ and whose sum is $b = - 127$ .

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Step 5.2.2.1.3.5.2.1

Factor $- 127$ out of $- 127 λ$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{18 λ^{2} - 127 (λ) + 182}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5.2.2

Rewrite $- 127$ as $- 36$ plus $- 91$

$p (λ) = (\frac{31}{6} - λ) (\frac{18 λ^{2} + (- 36 - 91) λ + 182}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5.2.3

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) (\frac{18 λ^{2} - 36 λ - 91 λ + 182}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5.3

Factor out the greatest common factor from each group.

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Step 5.2.2.1.3.5.3.1

Group the first two terms and the last two terms.

$p (λ) = (\frac{31}{6} - λ) (\frac{(18 λ^{2} - 36 λ) - 91 λ + 182}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5.3.2

Factor out the greatest common factor (GCF) from each group.

$p (λ) = (\frac{31}{6} - λ) (\frac{18 λ (λ - 2) - 91 (λ - 2)}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.3.5.4

Factor the polynomial by factoring out the greatest common factor, $λ - 2$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91)}{18} - \frac{16}{9} \cdot \frac{16}{9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.4

Multiply $- \frac{16}{9} \cdot \frac{16}{9}$ .

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Step 5.2.2.1.4.1

Multiply $\frac{16}{9}$ by $\frac{16}{9}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91)}{18} - \frac{16 \cdot 16}{9 \cdot 9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.4.2

Multiply $16$ by $16$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91)}{18} - \frac{256}{9 \cdot 9}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.1.4.3

Multiply $9$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91)}{18} - \frac{256}{81}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.2

To write $\frac{(λ - 2) (18 λ - 91)}{18}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91)}{18} \cdot \frac{9}{9} - \frac{256}{81}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.3

To write $- \frac{256}{81}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91)}{18} \cdot \frac{9}{9} - \frac{256}{81} \cdot \frac{2}{2}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.4

Write each expression with a common denominator of $162$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.4.1

Multiply $\frac{(λ - 2) (18 λ - 91)}{18}$ by $\frac{9}{9}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91) \cdot 9}{18 \cdot 9} - \frac{256}{81} \cdot \frac{2}{2}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.4.2

Multiply $18$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91) \cdot 9}{162} - \frac{256}{81} \cdot \frac{2}{2}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.4.3

Multiply $\frac{256}{81}$ by $\frac{2}{2}$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91) \cdot 9}{162} - \frac{256 \cdot 2}{81 \cdot 2}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.4.4

Multiply $81$ by $2$ .

$p (λ) = (\frac{31}{6} - λ) (\frac{(λ - 2) (18 λ - 91) \cdot 9}{162} - \frac{256 \cdot 2}{162}) + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.5

Combine the numerators over the common denominator.

$p (λ) = (\frac{31}{6} - λ) \frac{(λ - 2) (18 λ - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6

Simplify the numerator.

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Step 5.2.2.6.1

Expand $(λ - 2) (18 λ - 91)$ using the FOIL Method.

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Step 5.2.2.6.1.1

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{(λ (18 λ - 91) - 2 (18 λ - 91)) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.1.2

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{(λ (18 λ) + λ \cdot - 91 - 2 (18 λ - 91)) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.1.3

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{(λ (18 λ) + λ \cdot - 91 - 2 (18 λ) - 2 \cdot - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2

Simplify and combine like terms.

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Step 5.2.2.6.2.1

Simplify each term.

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Step 5.2.2.6.2.1.1

Rewrite using the commutative property of multiplication.

$p (λ) = (\frac{31}{6} - λ) \frac{(18 λ \cdot λ + λ \cdot - 91 - 2 (18 λ) - 2 \cdot - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2.1.2

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

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Step 5.2.2.6.2.1.2.1

Move $λ$ .

$p (λ) = (\frac{31}{6} - λ) \frac{(18 (λ \cdot λ) + λ \cdot - 91 - 2 (18 λ) - 2 \cdot - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2.1.2.2

Multiply $λ$ by $λ$ .

$p (λ) = (\frac{31}{6} - λ) \frac{(18 λ^{2} + λ \cdot - 91 - 2 (18 λ) - 2 \cdot - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2.1.3

Move $- 91$ to the left of $λ$ .

$p (λ) = (\frac{31}{6} - λ) \frac{(18 λ^{2} - 91 \cdot λ - 2 (18 λ) - 2 \cdot - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2.1.4

Multiply $18$ by $- 2$ .

$p (λ) = (\frac{31}{6} - λ) \frac{(18 λ^{2} - 91 λ - 36 λ - 2 \cdot - 91) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2.1.5

Multiply $- 2$ by $- 91$ .

$p (λ) = (\frac{31}{6} - λ) \frac{(18 λ^{2} - 91 λ - 36 λ + 182) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.2.2

Subtract $36 λ$ from $- 91 λ$ .

$p (λ) = (\frac{31}{6} - λ) \frac{(18 λ^{2} - 127 λ + 182) \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.3

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{18 λ^{2} \cdot 9 - 127 λ \cdot 9 + 182 \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.4

Simplify.

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Step 5.2.2.6.4.1

Multiply $9$ by $18$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 127 λ \cdot 9 + 182 \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.4.2

Multiply $9$ by $- 127$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 182 \cdot 9 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.4.3

Multiply $182$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1638 - 256 \cdot 2}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.5

Multiply $- 256$ by $2$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1638 - 512}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.2.2.6.6

Subtract $512$ from $1638$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} | \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} | - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3

Evaluate $| \begin{array}{c} - \frac{73}{18} & \frac{16}{9} \\ - \frac{16}{9} & 2 - λ \end{array} |$ .

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Step 5.3.1

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

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{18} (2 - λ) - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

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

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{18} \cdot 2 - \frac{73}{18} (- λ) - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.2.1

Move the leading negative in $- \frac{73}{18}$ into the numerator.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{18} \cdot 2 - \frac{73}{18} (- λ) - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.2.2

Factor $2$ out of $18$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{2 (9)} \cdot 2 - \frac{73}{18} (- λ) - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.2.3

Cancel the common factor.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{2 \cdot 9} \cdot 2 - \frac{73}{18} (- λ) - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.2.4

Rewrite the expression.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{9} - \frac{73}{18} (- λ) - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.3

Multiply $- \frac{73}{18} (- λ)$ .

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Step 5.3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{9} + 1 (\frac{73}{18}) λ - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.3.2

Multiply $\frac{73}{18}$ by $1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{9} + \frac{73}{18} λ - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.3.3

Combine $\frac{73}{18}$ and $λ$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{- 73}{9} + \frac{73 λ}{18} - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.4

Move the negative in front of the fraction.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{9} + \frac{73 λ}{18} - (- \frac{16}{9} \cdot \frac{16}{9})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.5

Multiply $- \frac{16}{9} \cdot \frac{16}{9}$ .

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Step 5.3.2.1.5.1

Multiply $\frac{16}{9}$ by $\frac{16}{9}$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{9} + \frac{73 λ}{18} - - \frac{16 \cdot 16}{9 \cdot 9}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.5.2

Multiply $16$ by $16$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{9} + \frac{73 λ}{18} - - \frac{256}{9 \cdot 9}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.5.3

Multiply $9$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{9} + \frac{73 λ}{18} - - \frac{256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.6

Multiply $- - \frac{256}{81}$ .

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Step 5.3.2.1.6.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{9} + \frac{73 λ}{18} + 1 (\frac{256}{81})) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.1.6.2

Multiply $\frac{256}{81}$ by $1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (- \frac{73}{9} + \frac{73 λ}{18} + \frac{256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.2

To write $- \frac{73}{9}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{73}{9} \cdot \frac{9}{9} + \frac{256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.3

Write each expression with a common denominator of $81$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.2.3.1

Multiply $\frac{73}{9}$ by $\frac{9}{9}$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{73 \cdot 9}{9 \cdot 9} + \frac{256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.3.2

Multiply $9$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{73 \cdot 9}{81} + \frac{256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.4

Combine the numerators over the common denominator.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} + \frac{- 73 \cdot 9 + 256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.5

Simplify the numerator.

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Step 5.3.2.5.1

Multiply $- 73$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} + \frac{- 657 + 256}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.5.2

Add $- 657$ and $256$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} + \frac{- 401}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.3.2.6

Move the negative in front of the fraction.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} | \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$

Step 5.4

Evaluate $| \begin{array}{c} - \frac{73}{18} & \frac{91}{18} - λ \\ - \frac{16}{9} & \frac{16}{9} \end{array} |$ .

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Step 5.4.1

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

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{73}{18} \cdot \frac{16}{9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

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

Cancel the common factor of $2$ .

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Step 5.4.2.1.1.1

Move the leading negative in $- \frac{73}{18}$ into the numerator.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 73}{18} \cdot \frac{16}{9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.1.2

Factor $2$ out of $18$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 73}{2 (9)} \cdot \frac{16}{9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.1.3

Factor $2$ out of $16$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 73}{2 \cdot 9} \cdot \frac{2 \cdot 8}{9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.1.4

Cancel the common factor.

Step 5.4.2.1.1.5

Rewrite the expression.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 73}{9} \cdot \frac{8}{9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.2

Multiply $\frac{- 73}{9}$ by $\frac{8}{9}$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 73 \cdot 8}{9 \cdot 9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.3

Multiply $- 73$ by $8$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 584}{9 \cdot 9} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.4

Multiply $9$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 584}{81} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.5

Move the negative in front of the fraction.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (- \frac{16}{9} (\frac{91}{18} - λ)))$

Step 5.4.2.1.6

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (- \frac{16}{9} \cdot \frac{91}{18} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.7

Cancel the common factor of $2$ .

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Step 5.4.2.1.7.1

Move the leading negative in $- \frac{16}{9}$ into the numerator.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 16}{9} \cdot \frac{91}{18} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.7.2

Factor $2$ out of $- 16$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{2 (- 8)}{9} \cdot \frac{91}{18} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.7.3

Factor $2$ out of $18$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{2 \cdot - 8}{9} \cdot \frac{91}{2 \cdot 9} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.7.4

Cancel the common factor.

Step 5.4.2.1.7.5

Rewrite the expression.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 8}{9} \cdot \frac{91}{9} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.8

Multiply $\frac{- 8}{9}$ by $\frac{91}{9}$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 8 \cdot 91}{9 \cdot 9} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.9

Multiply $- 8$ by $91$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 728}{9 \cdot 9} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.10

Multiply $9$ by $9$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 728}{81} - \frac{16}{9} (- λ)))$

Step 5.4.2.1.11

Multiply $- \frac{16}{9} (- λ)$ .

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Step 5.4.2.1.11.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 728}{81} + 1 (\frac{16}{9}) λ))$

Step 5.4.2.1.11.2

Multiply $\frac{16}{9}$ by $1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 728}{81} + \frac{16}{9} λ))$

Step 5.4.2.1.11.3

Combine $\frac{16}{9}$ and $λ$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (\frac{- 728}{81} + \frac{16 λ}{9}))$

Step 5.4.2.1.12

Move the negative in front of the fraction.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - (- \frac{728}{81} + \frac{16 λ}{9}))$

Step 5.4.2.1.13

Apply the distributive property.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} - - \frac{728}{81} - \frac{16 λ}{9})$

Step 5.4.2.1.14

Multiply $- - \frac{728}{81}$ .

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Step 5.4.2.1.14.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} + 1 (\frac{728}{81}) - \frac{16 λ}{9})$

Step 5.4.2.1.14.2

Multiply $\frac{728}{81}$ by $1$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{584}{81} + \frac{728}{81} - \frac{16 λ}{9})$

Step 5.4.2.2

Combine the numerators over the common denominator.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{- 584 + 728}{81} + \frac{- 16 λ}{9})$

Step 5.4.2.3

Add $- 584$ and $728$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{144}{81} + \frac{- 16 λ}{9})$

Step 5.4.2.4

Simplify each term.

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Step 5.4.2.4.1

Cancel the common factor of $144$ and $81$ .

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Step 5.4.2.4.1.1

Factor $9$ out of $144$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{9 (16)}{81} + \frac{- 16 λ}{9})$

Step 5.4.2.4.1.2

Cancel the common factors.

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Step 5.4.2.4.1.2.1

Factor $9$ out of $81$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{9 \cdot 16}{9 \cdot 9} + \frac{- 16 λ}{9})$

Step 5.4.2.4.1.2.2

Cancel the common factor.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{9 \cdot 16}{9 \cdot 9} + \frac{- 16 λ}{9})$

Step 5.4.2.4.1.2.3

Rewrite the expression.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{16}{9} + \frac{- 16 λ}{9})$

Step 5.4.2.4.2

Move the negative in front of the fraction.

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (\frac{16}{9} - \frac{16 λ}{9})$

Step 5.4.2.5

Reorder $\frac{16}{9}$ and $- \frac{16 λ}{9}$ .

$p (λ) = (\frac{31}{6} - λ) \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

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

Apply the distributive property.

$p (λ) = \frac{31}{6} \cdot \frac{162 λ^{2} - 1143 λ + 1126}{162} - λ \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.2

Multiply $\frac{31}{6} \cdot \frac{162 λ^{2} - 1143 λ + 1126}{162}$ .

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Step 5.5.1.2.1

Multiply $\frac{31}{6}$ by $\frac{162 λ^{2} - 1143 λ + 1126}{162}$ .

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126)}{6 \cdot 162} - λ \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.2.2

Multiply $6$ by $162$ .

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126)}{972} - λ \frac{162 λ^{2} - 1143 λ + 1126}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.3

Combine $\frac{162 λ^{2} - 1143 λ + 1126}{162}$ and $λ$ .

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126)}{972} - \frac{(162 λ^{2} - 1143 λ + 1126) λ}{162} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.4

To write $- \frac{(162 λ^{2} - 1143 λ + 1126) λ}{162}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126)}{972} - \frac{(162 λ^{2} - 1143 λ + 1126) λ}{162} \cdot \frac{6}{6} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.5

Write each expression with a common denominator of $972$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.1.5.1

Multiply $\frac{(162 λ^{2} - 1143 λ + 1126) λ}{162}$ by $\frac{6}{6}$ .

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126)}{972} - \frac{(162 λ^{2} - 1143 λ + 1126) λ \cdot 6}{162 \cdot 6} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.5.2

Multiply $162$ by $6$ .

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126)}{972} - \frac{(162 λ^{2} - 1143 λ + 1126) λ \cdot 6}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.6

Combine the numerators over the common denominator.

$p (λ) = \frac{31 (162 λ^{2} - 1143 λ + 1126) - (162 λ^{2} - 1143 λ + 1126) λ \cdot 6}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.7

Simplify the numerator.

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Step 5.5.1.7.1

Factor $162 λ^{2} - 1143 λ + 1126$ out of $31 (162 λ^{2} - 1143 λ + 1126) - (162 λ^{2} - 1143 λ + 1126) λ \cdot 6$ .

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Step 5.5.1.7.1.1

Factor $162 λ^{2} - 1143 λ + 1126$ out of $31 (162 λ^{2} - 1143 λ + 1126)$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) \cdot 31 - (162 λ^{2} - 1143 λ + 1126) λ \cdot 6}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.7.1.2

Factor $162 λ^{2} - 1143 λ + 1126$ out of $- (162 λ^{2} - 1143 λ + 1126) λ \cdot 6$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) \cdot 31 + (162 λ^{2} - 1143 λ + 1126) (- 1 λ \cdot 6)}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.7.1.3

Factor $162 λ^{2} - 1143 λ + 1126$ out of $(162 λ^{2} - 1143 λ + 1126) \cdot 31 + (162 λ^{2} - 1143 λ + 1126) (- 1 λ \cdot 6)$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 1 λ \cdot 6)}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.7.2

Rewrite $- 1 λ$ as $- λ$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - λ \cdot 6)}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.7.3

Multiply $6$ by $- 1$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{73}{18} (\frac{73 λ}{18} - \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.8

Apply the distributive property.

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{73}{18} \cdot \frac{73 λ}{18} + \frac{73}{18} (- \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.9

Multiply $\frac{73}{18} \cdot \frac{73 λ}{18}$ .

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Step 5.5.1.9.1

Multiply $\frac{73}{18}$ by $\frac{73 λ}{18}$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{73 (73 λ)}{18 \cdot 18} + \frac{73}{18} (- \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.9.2

Multiply $73$ by $73$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{18 \cdot 18} + \frac{73}{18} (- \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.9.3

Multiply $18$ by $18$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} + \frac{73}{18} (- \frac{401}{81}) - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.10

Multiply $\frac{73}{18} (- \frac{401}{81})$ .

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Step 5.5.1.10.1

Multiply $\frac{73}{18}$ by $\frac{401}{81}$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{73 \cdot 401}{18 \cdot 81} - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.10.2

Multiply $73$ by $401$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{18 \cdot 81} - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.10.3

Multiply $18$ by $81$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} - \frac{16}{9} (- \frac{16 λ}{9} + \frac{16}{9})$

Step 5.5.1.11

Apply the distributive property.

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} - \frac{16}{9} (- \frac{16 λ}{9}) - \frac{16}{9} \cdot \frac{16}{9}$

Step 5.5.1.12

Multiply $- \frac{16}{9} (- \frac{16 λ}{9})$ .

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Step 5.5.1.12.1

Multiply $- 1$ by $- 1$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + 1 (\frac{16}{9}) \frac{16 λ}{9} - \frac{16}{9} \cdot \frac{16}{9}$

Step 5.5.1.12.2

Multiply $\frac{16}{9}$ by $1$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{16}{9} \cdot \frac{16 λ}{9} - \frac{16}{9} \cdot \frac{16}{9}$

Step 5.5.1.12.3

Multiply $\frac{16}{9}$ by $\frac{16 λ}{9}$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{16 (16 λ)}{9 \cdot 9} - \frac{16}{9} \cdot \frac{16}{9}$

Step 5.5.1.12.4

Multiply $16$ by $16$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{256 λ}{9 \cdot 9} - \frac{16}{9} \cdot \frac{16}{9}$

Step 5.5.1.12.5

Multiply $9$ by $9$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{16}{9} \cdot \frac{16}{9}$

Step 5.5.1.13

Multiply $- \frac{16}{9} \cdot \frac{16}{9}$ .

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Step 5.5.1.13.1

Multiply $\frac{16}{9}$ by $\frac{16}{9}$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{16 \cdot 16}{9 \cdot 9}$

Step 5.5.1.13.2

Multiply $16$ by $16$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{9 \cdot 9}$

Step 5.5.1.13.3

Multiply $9$ by $9$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.2

To write $\frac{5329 λ}{324}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ}{324} \cdot \frac{3}{3} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.3

Write each expression with a common denominator of $972$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.1

Multiply $\frac{5329 λ}{324}$ by $\frac{3}{3}$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ \cdot 3}{324 \cdot 3} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.3.2

Multiply $324$ by $3$ .

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)}{972} + \frac{5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.4

Combine the numerators over the common denominator.

$p (λ) = \frac{(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5

Simplify each term.

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Step 5.5.5.1

Simplify the numerator.

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Step 5.5.5.1.1

Expand $(162 λ^{2} - 1143 λ + 1126) (31 - 6 λ)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = \frac{162 λ^{2} \cdot 31 + 162 λ^{2} (- 6 λ) - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2

Simplify each term.

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Step 5.5.5.1.2.1

Multiply $31$ by $162$ .

$p (λ) = \frac{5022 λ^{2} + 162 λ^{2} (- 6 λ) - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.2

Rewrite using the commutative property of multiplication.

$p (λ) = \frac{5022 λ^{2} + 162 \cdot - 6 λ^{2} λ - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.3

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

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Step 5.5.5.1.2.3.1

Move $λ$ .

$p (λ) = \frac{5022 λ^{2} + 162 \cdot - 6 (λ \cdot λ^{2}) - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.3.2

Multiply $λ$ by $λ^{2}$ .

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Step 5.5.5.1.2.3.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = \frac{5022 λ^{2} + 162 \cdot - 6 (λ^{1} λ^{2}) - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.3.2.2

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

$p (λ) = \frac{5022 λ^{2} + 162 \cdot - 6 λ^{1 + 2} - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.3.3

Add $1$ and $2$ .

$p (λ) = \frac{5022 λ^{2} + 162 \cdot - 6 λ^{3} - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.4

Multiply $162$ by $- 6$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 1143 λ \cdot 31 - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.5

Multiply $31$ by $- 1143$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ - 1143 λ (- 6 λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.6

Rewrite using the commutative property of multiplication.

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ - 1143 \cdot - 6 λ \cdot λ + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.7

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

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Step 5.5.5.1.2.7.1

Move $λ$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ - 1143 \cdot - 6 (λ \cdot λ) + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.7.2

Multiply $λ$ by $λ$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ - 1143 \cdot - 6 λ^{2} + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.8

Multiply $- 1143$ by $- 6$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ + 6858 λ^{2} + 1126 \cdot 31 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.9

Multiply $1126$ by $31$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ + 6858 λ^{2} + 34906 + 1126 (- 6 λ) + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.2.10

Multiply $- 6$ by $1126$ .

$p (λ) = \frac{5022 λ^{2} - 972 λ^{3} - 35433 λ + 6858 λ^{2} + 34906 - 6756 λ + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.3

Add $5022 λ^{2}$ and $6858 λ^{2}$ .

$p (λ) = \frac{11880 λ^{2} - 972 λ^{3} - 35433 λ + 34906 - 6756 λ + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.4

Subtract $6756 λ$ from $- 35433 λ$ .

$p (λ) = \frac{11880 λ^{2} - 972 λ^{3} - 42189 λ + 34906 + 5329 λ \cdot 3}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.5

Multiply $3$ by $5329$ .

$p (λ) = \frac{11880 λ^{2} - 972 λ^{3} - 42189 λ + 34906 + 15987 λ}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.6

Add $- 42189 λ$ and $15987 λ$ .

$p (λ) = \frac{11880 λ^{2} - 972 λ^{3} - 26202 λ + 34906}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.7

Reorder terms.

$p (λ) = \frac{- 972 λ^{3} + 11880 λ^{2} - 26202 λ + 34906}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8

Factor $2$ out of $- 972 λ^{3} + 11880 λ^{2} - 26202 λ + 34906$ .

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Step 5.5.5.1.8.1

Factor $2$ out of $- 972 λ^{3}$ .

$p (λ) = \frac{2 (- 486 λ^{3}) + 11880 λ^{2} - 26202 λ + 34906}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8.2

Factor $2$ out of $11880 λ^{2}$ .

$p (λ) = \frac{2 (- 486 λ^{3}) + 2 (5940 λ^{2}) - 26202 λ + 34906}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8.3

Factor $2$ out of $- 26202 λ$ .

$p (λ) = \frac{2 (- 486 λ^{3}) + 2 (5940 λ^{2}) + 2 (- 13101 λ) + 34906}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8.4

Factor $2$ out of $34906$ .

$p (λ) = \frac{2 (- 486 λ^{3}) + 2 (5940 λ^{2}) + 2 (- 13101 λ) + 2 (17453)}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8.5

Factor $2$ out of $2 (- 486 λ^{3}) + 2 (5940 λ^{2})$ .

$p (λ) = \frac{2 (- 486 λ^{3} + 5940 λ^{2}) + 2 (- 13101 λ) + 2 (17453)}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8.6

Factor $2$ out of $2 (- 486 λ^{3} + 5940 λ^{2}) + 2 (- 13101 λ)$ .

$p (λ) = \frac{2 (- 486 λ^{3} + 5940 λ^{2} - 13101 λ) + 2 (17453)}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.1.8.7

Factor $2$ out of $2 (- 486 λ^{3} + 5940 λ^{2} - 13101 λ) + 2 (17453)$ .

$p (λ) = \frac{2 (- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453)}{972} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.2

Cancel the common factors.

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Step 5.5.5.2.1

Factor $2$ out of $972$ .

$p (λ) = \frac{2 (- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453)}{2 \cdot 486} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.2.2

Cancel the common factor.

$p (λ) = \frac{2 (- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453)}{2 \cdot 486} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.5.2.3

Rewrite the expression.

$p (λ) = \frac{- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453}{486} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.6

To write $\frac{- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453}{486}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$p (λ) = \frac{- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453}{486} \cdot \frac{3}{3} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.7

Write each expression with a common denominator of $1458$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.7.1

Multiply $\frac{- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453}{486}$ by $\frac{3}{3}$ .

$p (λ) = \frac{(- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453) \cdot 3}{486 \cdot 3} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.7.2

Multiply $486$ by $3$ .

$p (λ) = \frac{(- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453) \cdot 3}{1458} - \frac{29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.8

Combine the numerators over the common denominator.

$p (λ) = \frac{(- 486 λ^{3} + 5940 λ^{2} - 13101 λ + 17453) \cdot 3 - 29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.9

Simplify the numerator.

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Step 5.5.9.1

Apply the distributive property.

$p (λ) = \frac{- 486 λ^{3} \cdot 3 + 5940 λ^{2} \cdot 3 - 13101 λ \cdot 3 + 17453 \cdot 3 - 29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.9.2

Simplify.

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Step 5.5.9.2.1

Multiply $3$ by $- 486$ .

$p (λ) = \frac{- 1458 λ^{3} + 5940 λ^{2} \cdot 3 - 13101 λ \cdot 3 + 17453 \cdot 3 - 29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.9.2.2

Multiply $3$ by $5940$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 13101 λ \cdot 3 + 17453 \cdot 3 - 29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.9.2.3

Multiply $3$ by $- 13101$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 17453 \cdot 3 - 29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.9.2.4

Multiply $17453$ by $3$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 52359 - 29273}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.9.3

Subtract $29273$ from $52359$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 23086}{1458} + \frac{256 λ}{81} - \frac{256}{81}$

Step 5.5.10

To write $\frac{256 λ}{81}$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 23086}{1458} + \frac{256 λ}{81} \cdot \frac{18}{18} - \frac{256}{81}$

Step 5.5.11

Write each expression with a common denominator of $1458$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.11.1

Multiply $\frac{256 λ}{81}$ by $\frac{18}{18}$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 23086}{1458} + \frac{256 λ \cdot 18}{81 \cdot 18} - \frac{256}{81}$

Step 5.5.11.2

Multiply $81$ by $18$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 23086}{1458} + \frac{256 λ \cdot 18}{1458} - \frac{256}{81}$

Step 5.5.12

Combine the numerators over the common denominator.

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 23086 + 256 λ \cdot 18}{1458} - \frac{256}{81}$

Step 5.5.13

Simplify the numerator.

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Step 5.5.13.1

Multiply $18$ by $256$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 39303 λ + 23086 + 4608 λ}{1458} - \frac{256}{81}$

Step 5.5.13.2

Add $- 39303 λ$ and $4608 λ$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 23086}{1458} - \frac{256}{81}$

Step 5.5.14

To write $- \frac{256}{81}$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 23086}{1458} - \frac{256}{81} \cdot \frac{18}{18}$

Step 5.5.15

Write each expression with a common denominator of $1458$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.15.1

Multiply $\frac{256}{81}$ by $\frac{18}{18}$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 23086}{1458} - \frac{256 \cdot 18}{81 \cdot 18}$

Step 5.5.15.2

Multiply $81$ by $18$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 23086}{1458} - \frac{256 \cdot 18}{1458}$

Step 5.5.16

Combine the numerators over the common denominator.

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 23086 - 256 \cdot 18}{1458}$

Step 5.5.17

Simplify the numerator.

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Step 5.5.17.1

Multiply $- 256$ by $18$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 23086 - 4608}{1458}$

Step 5.5.17.2

Subtract $4608$ from $23086$ .

$p (λ) = \frac{- 1458 λ^{3} + 17820 λ^{2} - 34695 λ + 18478}{1458}$

Step 5.5.18

Factor $- 1$ out of $- 1458 λ^{3}$ .

$p (λ) = \frac{- (1458 λ^{3}) + 17820 λ^{2} - 34695 λ + 18478}{1458}$

Step 5.5.19

Factor $- 1$ out of $17820 λ^{2}$ .

$p (λ) = \frac{- (1458 λ^{3}) - (- 17820 λ^{2}) - 34695 λ + 18478}{1458}$

Step 5.5.20

Factor $- 1$ out of $- (1458 λ^{3}) - (- 17820 λ^{2})$ .

$p (λ) = \frac{- (1458 λ^{3} - 17820 λ^{2}) - 34695 λ + 18478}{1458}$

Step 5.5.21

Factor $- 1$ out of $- 34695 λ$ .

$p (λ) = \frac{- (1458 λ^{3} - 17820 λ^{2}) - (34695 λ) + 18478}{1458}$

Step 5.5.22

Factor $- 1$ out of $- (1458 λ^{3} - 17820 λ^{2}) - (34695 λ)$ .

$p (λ) = \frac{- (1458 λ^{3} - 17820 λ^{2} + 34695 λ) + 18478}{1458}$

Step 5.5.23

Rewrite $18478$ as $- 1 (- 18478)$ .

$p (λ) = \frac{- (1458 λ^{3} - 17820 λ^{2} + 34695 λ) - 1 (- 18478)}{1458}$

Step 5.5.24

Factor $- 1$ out of $- (1458 λ^{3} - 17820 λ^{2} + 34695 λ) - 1 (- 18478)$ .

$p (λ) = \frac{- (1458 λ^{3} - 17820 λ^{2} + 34695 λ - 18478)}{1458}$

Step 5.5.25

Rewrite $- (1458 λ^{3} - 17820 λ^{2} + 34695 λ - 18478)$ as $- 1 (1458 λ^{3} - 17820 λ^{2} + 34695 λ - 18478)$ .

$p (λ) = \frac{- 1 (1458 λ^{3} - 17820 λ^{2} + 34695 λ - 18478)}{1458}$

Step 5.5.26

Move the negative in front of the fraction.

$p (λ) = - \frac{1458 λ^{3} - 17820 λ^{2} + 34695 λ - 18478}{1458}$

Step 6

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

$- \frac{1458 λ^{3} - 17820 λ^{2} + 34695 λ - 18478}{1458} = 0$

Step 7

Solve for $λ$ .

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Step 7.1

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

$λ \approx 1.05340571, 1.20780975, 9.96100674$