Linear Algebra Examples

[\begin{matrix} 9 & 9 7 & 8 \end{matrix}]

$[\begin{array}{c} 9 & 9 \\ 7 & 8 \end{array}]$

Step 1

Set up the formula to find the characteristic equation

p (λ)

$p (λ)$ .

p (λ) = determinant (A - λ I_{2})

$p (λ) = determinant (A - λ I_{2})$

Step 2

The identity matrix or unit matrix of size

2

$2$ is the

2 \times 2

$2 \times 2$ square matrix with ones on the main diagonal and zeros elsewhere.

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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

Step 3

Substitute the known values into

p (λ) = determinant (A - λ I_{2})

$p (λ) = determinant (A - λ I_{2})$ .

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Step 3.1

Substitute

[\begin{matrix} 9 & 9 7 & 8 \end{matrix}]

$[\begin{array}{c} 9 & 9 \\ 7 & 8 \end{array}]$ for

A

$A$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] - λ I_{2})

$p (λ) = determinant ([\begin{array}{c} 9 & 9 \\ 7 & 8 \end{array}] - λ I_{2})$

Step 3.2

Substitute

[\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$ for

I_{2}

$I_{2}$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] - λ [\begin{matrix} 1 & 0 0 & 1 \end{matrix}])

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

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] - λ [\begin{matrix} 1 & 0 0 & 1 \end{matrix}])

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply

- λ

$- λ$ by each element of the matrix.

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ \cdot 1 & - λ \cdot 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

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

Step 4.1.2

Simplify each element in the matrix.

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Step 4.1.2.1

Multiply

- 1

$- 1$ by

1

$1$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & - λ \cdot 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

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

Step 4.1.2.2

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 4.1.2.2.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 λ - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

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

Step 4.1.2.2.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

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

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

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

Step 4.1.2.3

Multiply

- λ \cdot 0

$- λ \cdot 0$ .

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Step 4.1.2.3.1

Multiply

0

$0$ by

- 1

$- 1$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 0 λ & - λ \cdot 1 \end{matrix}])

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

Step 4.1.2.3.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \cdot 1 \end{matrix}])

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

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \cdot 1 \end{matrix}])

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

Step 4.1.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 9 & 9 \\ 7 & 8 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 9 & 9 \\ 7 & 8 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

p (λ) = determinant ([\begin{matrix} 9 & 9 7 & 8 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 9 & 9 \\ 7 & 8 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

p (λ) = determinant [\begin{matrix} 9 - λ & 9 + 0 7 + 0 & 8 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 9 - λ & 9 + 0 \\ 7 + 0 & 8 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add

9

$9$ and

0

$0$ .

p (λ) = determinant [\begin{matrix} 9 - λ & 9 7 + 0 & 8 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 9 - λ & 9 \\ 7 + 0 & 8 - λ \end{array}]$

Step 4.3.2

Add

7

$7$ and

0

$0$ .

p (λ) = determinant [\begin{matrix} 9 - λ & 9 7 & 8 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 9 - λ & 9 \\ 7 & 8 - λ \end{array}]$

p (λ) = determinant [\begin{matrix} 9 - λ & 9 7 & 8 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 9 - λ & 9 \\ 7 & 8 - λ \end{array}]$

p (λ) = determinant [\begin{matrix} 9 - λ & 9 7 & 8 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 9 - λ & 9 \\ 7 & 8 - λ \end{array}]$

Step 5

Find the determinant.

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Step 5.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

p (λ) = (9 - λ) (8 - λ) - 7 \cdot 9

$p (λ) = (9 - λ) (8 - λ) - 7 \cdot 9$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Expand

(9 - λ) (8 - λ)

$(9 - λ) (8 - λ)$ using the FOIL Method.

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Step 5.2.1.1.1

Apply the distributive property.

p (λ) = 9 (8 - λ) - λ (8 - λ) - 7 \cdot 9

$p (λ) = 9 (8 - λ) - λ (8 - λ) - 7 \cdot 9$

Step 5.2.1.1.2

Apply the distributive property.

p (λ) = 9 \cdot 8 + 9 (- λ) - λ (8 - λ) - 7 \cdot 9

$p (λ) = 9 \cdot 8 + 9 (- λ) - λ (8 - λ) - 7 \cdot 9$

Step 5.2.1.1.3

Apply the distributive property.

p (λ) = 9 \cdot 8 + 9 (- λ) - λ \cdot 8 - λ (- λ) - 7 \cdot 9

$p (λ) = 9 \cdot 8 + 9 (- λ) - λ \cdot 8 - λ (- λ) - 7 \cdot 9$

p (λ) = 9 \cdot 8 + 9 (- λ) - λ \cdot 8 - λ (- λ) - 7 \cdot 9

$p (λ) = 9 \cdot 8 + 9 (- λ) - λ \cdot 8 - λ (- λ) - 7 \cdot 9$

Step 5.2.1.2

Simplify and combine like terms.

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Step 5.2.1.2.1

Simplify each term.

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Step 5.2.1.2.1.1

Multiply

9

$9$ by

8

$8$ .

p (λ) = 72 + 9 (- λ) - λ \cdot 8 - λ (- λ) - 7 \cdot 9

$p (λ) = 72 + 9 (- λ) - λ \cdot 8 - λ (- λ) - 7 \cdot 9$

Step 5.2.1.2.1.2

Multiply

- 1

$- 1$ by

9

$9$ .

p (λ) = 72 - 9 λ - λ \cdot 8 - λ (- λ) - 7 \cdot 9

$p (λ) = 72 - 9 λ - λ \cdot 8 - λ (- λ) - 7 \cdot 9$

Step 5.2.1.2.1.3

Multiply

8

$8$ by

- 1

$- 1$ .

p (λ) = 72 - 9 λ - 8 λ - λ (- λ) - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ - λ (- λ) - 7 \cdot 9$

Step 5.2.1.2.1.4

Rewrite using the commutative property of multiplication.

p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 λ \cdot λ - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 λ \cdot λ - 7 \cdot 9$

Step 5.2.1.2.1.5

Multiply

λ

$λ$ by

λ

$λ$ by adding the exponents.

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Step 5.2.1.2.1.5.1

Move

λ

$λ$ .

p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 (λ \cdot λ) - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 (λ \cdot λ) - 7 \cdot 9$

Step 5.2.1.2.1.5.2

Multiply

λ

$λ$ by

λ

$λ$ .

p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 λ^{2} - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 λ^{2} - 7 \cdot 9$

p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 λ^{2} - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ - 1 \cdot - 1 λ^{2} - 7 \cdot 9$

Step 5.2.1.2.1.6

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

p (λ) = 72 - 9 λ - 8 λ + 1 λ^{2} - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ + 1 λ^{2} - 7 \cdot 9$

Step 5.2.1.2.1.7

Multiply

λ^{2}

$λ^{2}$ by

1

$1$ .

p (λ) = 72 - 9 λ - 8 λ + λ^{2} - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ + λ^{2} - 7 \cdot 9$

p (λ) = 72 - 9 λ - 8 λ + λ^{2} - 7 \cdot 9

$p (λ) = 72 - 9 λ - 8 λ + λ^{2} - 7 \cdot 9$

Step 5.2.1.2.2

Subtract

8 λ

$8 λ$ from

- 9 λ

$- 9 λ$ .

p (λ) = 72 - 17 λ + λ^{2} - 7 \cdot 9

$p (λ) = 72 - 17 λ + λ^{2} - 7 \cdot 9$

p (λ) = 72 - 17 λ + λ^{2} - 7 \cdot 9

$p (λ) = 72 - 17 λ + λ^{2} - 7 \cdot 9$

Step 5.2.1.3

Multiply

- 7

$- 7$ by

9

$9$ .

p (λ) = 72 - 17 λ + λ^{2} - 63

$p (λ) = 72 - 17 λ + λ^{2} - 63$

p (λ) = 72 - 17 λ + λ^{2} - 63

$p (λ) = 72 - 17 λ + λ^{2} - 63$

Step 5.2.2

Subtract

63

$63$ from

72

$72$ .

p (λ) = - 17 λ + λ^{2} + 9

$p (λ) = - 17 λ + λ^{2} + 9$

Step 5.2.3

Reorder

- 17 λ

$- 17 λ$ and

λ^{2}

$λ^{2}$ .

p (λ) = λ^{2} - 17 λ + 9

$p (λ) = λ^{2} - 17 λ + 9$

p (λ) = λ^{2} - 17 λ + 9

$p (λ) = λ^{2} - 17 λ + 9$

p (λ) = λ^{2} - 17 λ + 9

$p (λ) = λ^{2} - 17 λ + 9$