Linear Algebra Examples

$[\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}]$

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 $[\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] - λ I_{3})$

Step 3.2

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

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

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.

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply $- λ \cdot 0$ .

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Step 4.1.2.6.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

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Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

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Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 40 & - 84 & 0 \\ 18 & - 38 & 0 \\ 36 & - 72 & - 2 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 + 0 & 0 + 0 \\ 18 + 0 & - 38 - λ & 0 + 0 \\ 36 + 0 & - 72 + 0 & - 2 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add $- 84$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 & 0 + 0 \\ 18 + 0 & - 38 - λ & 0 + 0 \\ 36 + 0 & - 72 + 0 & - 2 - λ \end{array}]$

Step 4.3.2

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 & 0 \\ 18 + 0 & - 38 - λ & 0 + 0 \\ 36 + 0 & - 72 + 0 & - 2 - λ \end{array}]$

Step 4.3.3

Add $18$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 & 0 \\ 18 & - 38 - λ & 0 + 0 \\ 36 + 0 & - 72 + 0 & - 2 - λ \end{array}]$

Step 4.3.4

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 & 0 \\ 18 & - 38 - λ & 0 \\ 36 + 0 & - 72 + 0 & - 2 - λ \end{array}]$

Step 4.3.5

Add $36$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 & 0 \\ 18 & - 38 - λ & 0 \\ 36 & - 72 + 0 & - 2 - λ \end{array}]$

Step 4.3.6

Add $- 72$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 40 - λ & - 84 & 0 \\ 18 & - 38 - λ & 0 \\ 36 & - 72 & - 2 - λ \end{array}]$

Step 5

Find the determinant.

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Step 5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $3$ 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 $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 18 & - 38 - λ \\ 36 & - 72 \end{array} |$

Step 5.1.4

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

$0 | \begin{array}{c} 18 & - 38 - λ \\ 36 & - 72 \end{array} |$

Step 5.1.5

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 40 - λ & - 84 \\ 36 & - 72 \end{array} |$

Step 5.1.6

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 40 - λ & - 84 \\ 36 & - 72 \end{array} |$

Step 5.1.7

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

$| \begin{array}{c} 40 - λ & - 84 \\ 18 & - 38 - λ \end{array} |$

Step 5.1.8

Multiply element $a_{33}$ by its cofactor.

$(- 2 - λ) | \begin{array}{c} 40 - λ & - 84 \\ 18 & - 38 - λ \end{array} |$

Step 5.1.9

Add the terms together.

$p (λ) = 0 | \begin{array}{c} 18 & - 38 - λ \\ 36 & - 72 \end{array} | + 0 | \begin{array}{c} 40 - λ & - 84 \\ 36 & - 72 \end{array} | + (- 2 - λ) | \begin{array}{c} 40 - λ & - 84 \\ 18 & - 38 - λ \end{array} |$

Step 5.2

Multiply $0$ by $| \begin{array}{c} 18 & - 38 - λ \\ 36 & - 72 \end{array} |$ .

$p (λ) = 0 + 0 | \begin{array}{c} 40 - λ & - 84 \\ 36 & - 72 \end{array} | + (- 2 - λ) | \begin{array}{c} 40 - λ & - 84 \\ 18 & - 38 - λ \end{array} |$

Step 5.3

Multiply $0$ by $| \begin{array}{c} 40 - λ & - 84 \\ 36 & - 72 \end{array} |$ .

$p (λ) = 0 + 0 + (- 2 - λ) | \begin{array}{c} 40 - λ & - 84 \\ 18 & - 38 - λ \end{array} |$

Step 5.4

Evaluate $| \begin{array}{c} 40 - λ & - 84 \\ 18 & - 38 - λ \end{array} |$ .

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Step 5.4.1

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

$p (λ) = 0 + 0 + (- 2 - λ) ((40 - λ) (- 38 - λ) - 18 \cdot - 84)$

Step 5.4.2

Simplify the determinant.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Expand $(40 - λ) (- 38 - λ)$ using the FOIL Method.

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Step 5.4.2.1.1.1

Apply the distributive property.

$p (λ) = 0 + 0 + (- 2 - λ) (40 (- 38 - λ) - λ (- 38 - λ) - 18 \cdot - 84)$

Step 5.4.2.1.1.2

Apply the distributive property.

$p (λ) = 0 + 0 + (- 2 - λ) (40 \cdot - 38 + 40 (- λ) - λ (- 38 - λ) - 18 \cdot - 84)$

Step 5.4.2.1.1.3

Apply the distributive property.

$p (λ) = 0 + 0 + (- 2 - λ) (40 \cdot - 38 + 40 (- λ) - λ \cdot - 38 - λ (- λ) - 18 \cdot - 84)$

Step 5.4.2.1.2

Simplify and combine like terms.

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Step 5.4.2.1.2.1

Simplify each term.

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Step 5.4.2.1.2.1.1

Multiply $40$ by $- 38$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 + 40 (- λ) - λ \cdot - 38 - λ (- λ) - 18 \cdot - 84)$

Step 5.4.2.1.2.1.2

Multiply $- 1$ by $40$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ - λ \cdot - 38 - λ (- λ) - 18 \cdot - 84)$

Step 5.4.2.1.2.1.3

Multiply $- 38$ by $- 1$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ + 38 λ - λ (- λ) - 18 \cdot - 84)$

Step 5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ + 38 λ - 1 \cdot - 1 λ \cdot λ - 18 \cdot - 84)$

Step 5.4.2.1.2.1.5

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

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Step 5.4.2.1.2.1.5.1

Move $λ$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ + 38 λ - 1 \cdot - 1 (λ \cdot λ) - 18 \cdot - 84)$

Step 5.4.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ + 38 λ - 1 \cdot - 1 λ^{2} - 18 \cdot - 84)$

Step 5.4.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ + 38 λ + 1 λ^{2} - 18 \cdot - 84)$

Step 5.4.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 40 λ + 38 λ + λ^{2} - 18 \cdot - 84)$

Step 5.4.2.1.2.2

Add $- 40 λ$ and $38 λ$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 2 λ + λ^{2} - 18 \cdot - 84)$

Step 5.4.2.1.3

Multiply $- 18$ by $- 84$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 1520 - 2 λ + λ^{2} + 1512)$

Step 5.4.2.2

Add $- 1520$ and $1512$ .

$p (λ) = 0 + 0 + (- 2 - λ) (- 2 λ + λ^{2} - 8)$

Step 5.4.2.3

Reorder $- 2 λ$ and $λ^{2}$ .

$p (λ) = 0 + 0 + (- 2 - λ) (λ^{2} - 2 λ - 8)$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Combine the opposite terms in $0 + 0 + (- 2 - λ) (λ^{2} - 2 λ - 8)$ .

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Step 5.5.1.1

Add $0$ and $0$ .

$p (λ) = 0 + (- 2 - λ) (λ^{2} - 2 λ - 8)$

Step 5.5.1.2

Add $0$ and $(- 2 - λ) (λ^{2} - 2 λ - 8)$ .

$p (λ) = (- 2 - λ) (λ^{2} - 2 λ - 8)$

Step 5.5.2

Expand $(- 2 - λ) (λ^{2} - 2 λ - 8)$ by multiplying each term in the first expression by each term in the second expression.

$p (λ) = - 2 λ^{2} - 2 (- 2 λ) - 2 \cdot - 8 - λ \cdot λ^{2} - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3

Simplify each term.

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Step 5.5.3.1

Multiply $- 2$ by $- 2$ .

$p (λ) = - 2 λ^{2} + 4 λ - 2 \cdot - 8 - λ \cdot λ^{2} - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3.2

Multiply $- 2$ by $- 8$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ \cdot λ^{2} - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3.3

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

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Step 5.5.3.3.1

Move $λ^{2}$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - (λ^{2} λ) - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3.3.2

Multiply $λ^{2}$ by $λ$ .

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Step 5.5.3.3.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - (λ^{2} λ^{1}) - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3.3.2.2

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

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{2 + 1} - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3.3.3

Add $2$ and $1$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{3} - λ (- 2 λ) - λ \cdot - 8$

Step 5.5.3.4

Rewrite using the commutative property of multiplication.

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{3} - 1 \cdot - 2 λ \cdot λ - λ \cdot - 8$

Step 5.5.3.5

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

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Step 5.5.3.5.1

Move $λ$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{3} - 1 \cdot - 2 (λ \cdot λ) - λ \cdot - 8$

Step 5.5.3.5.2

Multiply $λ$ by $λ$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{3} - 1 \cdot - 2 λ^{2} - λ \cdot - 8$

Step 5.5.3.6

Multiply $- 1$ by $- 2$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{3} + 2 λ^{2} - λ \cdot - 8$

Step 5.5.3.7

Multiply $- 8$ by $- 1$ .

$p (λ) = - 2 λ^{2} + 4 λ + 16 - λ^{3} + 2 λ^{2} + 8 λ$

Step 5.5.4

Combine the opposite terms in $- 2 λ^{2} + 4 λ + 16 - λ^{3} + 2 λ^{2} + 8 λ$ .

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Step 5.5.4.1

Add $- 2 λ^{2}$ and $2 λ^{2}$ .

$p (λ) = 4 λ + 16 - λ^{3} + 0 + 8 λ$

Step 5.5.4.2

Add $4 λ + 16 - λ^{3}$ and $0$ .

$p (λ) = 4 λ + 16 - λ^{3} + 8 λ$

Step 5.5.5

Add $4 λ$ and $8 λ$ .

$p (λ) = 12 λ + 16 - λ^{3}$

Step 5.5.6

Move $16$ .

$p (λ) = 12 λ - λ^{3} + 16$

Step 5.5.7

Reorder $12 λ$ and $- λ^{3}$ .

$p (λ) = - λ^{3} + 12 λ + 16$