Linear Algebra Examples

A = [\begin{matrix} 2 & 1 4 & 0 \end{matrix}]

$A = [\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}]$

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} 2 & 1 4 & 0 \end{matrix}]

$[\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}]$ for

A

$A$ .

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] - λ I_{2})

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] - λ I_{2})$

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} 2 & 1 4 & 0 \end{matrix}] - λ [\begin{matrix} 1 & 0 0 & 1 \end{matrix}])

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

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] - λ [\begin{matrix} 1 & 0 0 & 1 \end{matrix}])

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

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} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ \cdot 1 & - λ \cdot 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

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} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & - λ \cdot 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

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} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 λ - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 - λ \cdot 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

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} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 0 λ & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply

0

$0$ by

λ

$λ$ .

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \cdot 1 \end{array}])$

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \cdot 1 \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

p (λ) = determinant ([\begin{matrix} 2 & 1 4 & 0 \end{matrix}] + [\begin{matrix} - λ & 0 0 & - λ \end{matrix}])

$p (λ) = determinant ([\begin{array}{c} 2 & 1 \\ 4 & 0 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

p (λ) = determinant [\begin{matrix} 2 - λ & 1 + 0 4 + 0 & 0 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 1 + 0 \\ 4 + 0 & 0 - λ \end{array}]$

Step 4.3

Simplify each element.

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Step 4.3.1

Add

1

$1$ and

0

$0$ .

p (λ) = determinant [\begin{matrix} 2 - λ & 1 4 + 0 & 0 - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 1 \\ 4 + 0 & 0 - λ \end{array}]$

Step 4.3.2

Add

4

$4$ and

0

$0$ .

p (λ) = determinant [\begin{matrix} 2 - λ & 1 4 & 0 - λ \end{matrix}]

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

Step 4.3.3

Subtract

λ

$λ$ from

0

$0$ .

p (λ) = determinant [\begin{matrix} 2 - λ & 1 4 & - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 1 \\ 4 & - λ \end{array}]$

p (λ) = determinant [\begin{matrix} 2 - λ & 1 4 & - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 1 \\ 4 & - λ \end{array}]$

p (λ) = determinant [\begin{matrix} 2 - λ & 1 4 & - λ \end{matrix}]

$p (λ) = determinant [\begin{array}{c} 2 - λ & 1 \\ 4 & - λ \end{array}]$

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) = (2 - λ) (- λ) - 4 \cdot 1

$p (λ) = (2 - λ) (- λ) - 4 \cdot 1$

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

Apply the distributive property.

p (λ) = 2 (- λ) - λ (- λ) - 4 \cdot 1

$p (λ) = 2 (- λ) - λ (- λ) - 4 \cdot 1$

Step 5.2.1.2

Multiply

- 1

$- 1$ by

2

$2$ .

p (λ) = - 2 λ - λ (- λ) - 4 \cdot 1

$p (λ) = - 2 λ - λ (- λ) - 4 \cdot 1$

Step 5.2.1.3

Rewrite using the commutative property of multiplication.

p (λ) = - 2 λ - 1 \cdot - 1 λ \cdot λ - 4 \cdot 1

$p (λ) = - 2 λ - 1 \cdot - 1 λ \cdot λ - 4 \cdot 1$

Step 5.2.1.4

Simplify each term.

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Step 5.2.1.4.1

Multiply

λ

$λ$ by

λ

$λ$ by adding the exponents.

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Step 5.2.1.4.1.1

Move

λ

$λ$ .

p (λ) = - 2 λ - 1 \cdot - 1 (λ \cdot λ) - 4 \cdot 1

$p (λ) = - 2 λ - 1 \cdot - 1 (λ \cdot λ) - 4 \cdot 1$

Step 5.2.1.4.1.2

Multiply

λ

$λ$ by

λ

$λ$ .

p (λ) = - 2 λ - 1 \cdot - 1 λ^{2} - 4 \cdot 1

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

p (λ) = - 2 λ - 1 \cdot - 1 λ^{2} - 4 \cdot 1

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

Step 5.2.1.4.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

p (λ) = - 2 λ + 1 λ^{2} - 4 \cdot 1

$p (λ) = - 2 λ + 1 λ^{2} - 4 \cdot 1$

Step 5.2.1.4.3

Multiply

λ^{2}

$λ^{2}$ by

1

$1$ .

p (λ) = - 2 λ + λ^{2} - 4 \cdot 1

$p (λ) = - 2 λ + λ^{2} - 4 \cdot 1$

p (λ) = - 2 λ + λ^{2} - 4 \cdot 1

$p (λ) = - 2 λ + λ^{2} - 4 \cdot 1$

Step 5.2.1.5

Multiply

- 4

$- 4$ by

1

$1$ .

p (λ) = - 2 λ + λ^{2} - 4

$p (λ) = - 2 λ + λ^{2} - 4$

Step 5.2.2

Reorder

$- 2 λ$ and

$λ^{2}$ .

$p (λ) = λ^{2} - 2 λ - 4$

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