Finite Math Examples

⎡ ⎢ ⎣ \begin{matrix} 0 & 0.1 & 0.9 0.6 & 0 & 0.4 0.9 & 0.1 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}]$

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

$p (λ) = determinant (A - λ I_{3})$ .

Tap for more steps...

Step 3.1

Substitute

$[\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}]$ for

$A$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] - λ I_{3})$

Step 3.2

Substitute

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

$I_{3}$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply

$- λ$ by each element of the matrix.

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

Tap for more steps...

Step 4.1.2.1

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply

$- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.2.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply

$- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.3.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply

$- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.4.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply

$- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.6.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply

$- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.7.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply

$- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.8.1

Multiply

$0$ by

$- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply

$0$ by

$λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply

$- 1$ by

$1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0.1 & 0.9 \\ 0.6 & 0 & 0.4 \\ 0.9 & 0.1 & 0 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 0 - λ & 0.1 + 0 & 0.9 + 0 \\ 0.6 + 0 & 0 - λ & 0.4 + 0 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3

Simplify each element.

Tap for more steps...

Step 4.3.1

Subtract

$λ$ from

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 + 0 & 0.9 + 0 \\ 0.6 + 0 & 0 - λ & 0.4 + 0 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.2

Add

$0.1$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 + 0 \\ 0.6 + 0 & 0 - λ & 0.4 + 0 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.3

Add

$0.9$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 + 0 & 0 - λ & 0.4 + 0 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.4

Add

$0.6$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 & 0 - λ & 0.4 + 0 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.5

Subtract

$λ$ from

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 & - λ & 0.4 + 0 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.6

Add

$0.4$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 & - λ & 0.4 \\ 0.9 + 0 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.7

Add

$0.9$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 & - λ & 0.4 \\ 0.9 & 0.1 + 0 & 0 - λ \end{array}]$

Step 4.3.8

Add

$0.1$ and

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 & - λ & 0.4 \\ 0.9 & 0.1 & 0 - λ \end{array}]$

Step 4.3.9

Subtract

$λ$ from

$0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0.1 & 0.9 \\ 0.6 & - λ & 0.4 \\ 0.9 & 0.1 & - λ \end{array}]$

Step 5

Find the determinant.

Tap for more steps...

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.

Tap for more steps...

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.

$| \begin{array}{c} - λ & 0.4 \\ 0.1 & - λ \end{array} |$

Step 5.1.4

Multiply element

$a_{11}$ by its cofactor.

$- λ | \begin{array}{c} - λ & 0.4 \\ 0.1 & - λ \end{array} |$

Step 5.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

$| \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} |$

Step 5.1.6

Multiply element

$a_{12}$ by its cofactor.

$- 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} |$

Step 5.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.1.8

Multiply element

$a_{13}$ by its cofactor.

$0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.1.9

Add the terms together.

$p (λ) = - λ | \begin{array}{c} - λ & 0.4 \\ 0.1 & - λ \end{array} | - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2

Evaluate

$| \begin{array}{c} - λ & 0.4 \\ 0.1 & - λ \end{array} |$ .

Tap for more steps...

Step 5.2.1

The determinant of a

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

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

$p (λ) = - λ (- λ (- λ) - 0.1 \cdot 0.4) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

Rewrite using the commutative property of multiplication.

$p (λ) = - λ (- 1 \cdot - 1 λ \cdot λ - 0.1 \cdot 0.4) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2.2.2

Multiply

$λ$ by

$λ$ by adding the exponents.

Tap for more steps...

Step 5.2.2.2.1

Move

$λ$ .

$p (λ) = - λ (- 1 \cdot - 1 (λ \cdot λ) - 0.1 \cdot 0.4) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2.2.2.2

Multiply

$λ$ by

$λ$ .

$p (λ) = - λ (- 1 \cdot - 1 λ^{2} - 0.1 \cdot 0.4) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2.2.3

Multiply

$- 1$ by

$- 1$ .

$p (λ) = - λ (1 λ^{2} - 0.1 \cdot 0.4) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2.2.4

Multiply

$λ^{2}$ by

$1$ .

$p (λ) = - λ (λ^{2} - 0.1 \cdot 0.4) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.2.2.5

Multiply

$- 0.1$ by

$0.4$ .

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 | \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} | + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.3

Evaluate

$| \begin{array}{c} 0.6 & 0.4 \\ 0.9 & - λ \end{array} |$ .

Tap for more steps...

Step 5.3.1

The determinant of a

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

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

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (0.6 (- λ) - 0.9 \cdot 0.4) + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.3.2

Simplify each term.

Tap for more steps...

Step 5.3.2.1

Multiply

$- 1$ by

$0.6$ .

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (- 0.6 λ - 0.9 \cdot 0.4) + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.3.2.2

Multiply

$- 0.9$ by

$0.4$ .

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (- 0.6 λ - 0.36) + 0.9 | \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$

Step 5.4

Evaluate

$| \begin{array}{c} 0.6 & - λ \\ 0.9 & 0.1 \end{array} |$ .

Tap for more steps...

Step 5.4.1

The determinant of a

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

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

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.6 \cdot 0.1 - 0.9 (- λ))$

Step 5.4.2

Simplify the determinant.

Tap for more steps...

Step 5.4.2.1

Simplify each term.

Tap for more steps...

Step 5.4.2.1.1

Multiply

$0.6$ by

$0.1$ .

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.06 - 0.9 (- λ))$

Step 5.4.2.1.2

Multiply

$- 1$ by

$- 0.9$ .

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.06 + 0.9 λ)$

Step 5.4.2.2

Reorder

$0.06$ and

$0.9 λ$ .

$p (λ) = - λ (λ^{2} - 0.04) - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5

Simplify the determinant.

Tap for more steps...

Step 5.5.1

Simplify each term.

Tap for more steps...

Step 5.5.1.1

Apply the distributive property.

$p (λ) = - λ \cdot λ^{2} - λ \cdot - 0.04 - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.2

Multiply

$λ$ by

$λ^{2}$ by adding the exponents.

Tap for more steps...

Step 5.5.1.2.1

Move

$λ^{2}$ .

$p (λ) = - (λ^{2} λ) - λ \cdot - 0.04 - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.2.2

Multiply

$λ^{2}$ by

$λ$ .

Tap for more steps...

Step 5.5.1.2.2.1

Raise

$λ$ to the power of

$1$ .

$p (λ) = - (λ^{2} λ^{1}) - λ \cdot - 0.04 - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - λ^{2 + 1} - λ \cdot - 0.04 - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.2.3

Add

$2$ and

$1$ .

$p (λ) = - λ^{3} - λ \cdot - 0.04 - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.3

Multiply

$- 0.04$ by

$- 1$ .

$p (λ) = - λ^{3} + 0.04 λ - 0.1 (- 0.6 λ - 0.36) + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.4

Apply the distributive property.

$p (λ) = - λ^{3} + 0.04 λ - 0.1 (- 0.6 λ) - 0.1 \cdot - 0.36 + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.5

Multiply

$- 0.6$ by

$- 0.1$ .

$p (λ) = - λ^{3} + 0.04 λ + 0.06 λ - 0.1 \cdot - 0.36 + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.6

Multiply

$- 0.1$ by

$- 0.36$ .

$p (λ) = - λ^{3} + 0.04 λ + 0.06 λ + 0.036 + 0.9 (0.9 λ + 0.06)$

Step 5.5.1.7

Apply the distributive property.

$p (λ) = - λ^{3} + 0.04 λ + 0.06 λ + 0.036 + 0.9 (0.9 λ) + 0.9 \cdot 0.06$

Step 5.5.1.8

Multiply

$0.9$ by

$0.9$ .

$p (λ) = - λ^{3} + 0.04 λ + 0.06 λ + 0.036 + 0.81 λ + 0.9 \cdot 0.06$

Step 5.5.1.9

Multiply

$0.9$ by

$0.06$ .

$p (λ) = - λ^{3} + 0.04 λ + 0.06 λ + 0.036 + 0.81 λ + 0.054$

Step 5.5.2

Add

$0.04 λ$ and

$0.06 λ$ .

$p (λ) = - λ^{3} + 0.1 λ + 0.036 + 0.81 λ + 0.054$

Step 5.5.3

Add

$0.1 λ$ and

$0.81 λ$ .

$p (λ) = - λ^{3} + 0.91 λ + 0.036 + 0.054$

Step 5.5.4

Add

$0.036$ and

$0.054$ .

$p (λ) = - λ^{3} + 0.91 λ + 0.09$

Step 6

Set the characteristic polynomial equal to

$0$ to find the eigenvalues

$λ$ .

$- λ^{3} + 0.91 λ + 0.09 = 0$

Step 7

Solve for

$λ$ .

Tap for more steps...

Step 7.1

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

$λ \approx - 0.9, - 0.1, 1$