Examples

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}]$

Step 1

Find the eigenvalues.

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Step 1.1

Set up the formula to find the characteristic equation $p (λ)$ .

$p (λ) = determinant (A - λ I_{2})$

Step 1.2

The identity matrix or unit matrix of size $2$ is the $2 \times 2$ square matrix with ones on the main diagonal and zeros elsewhere.

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

Step 1.3

Substitute the known values into $p (λ) = determinant (A - λ I_{2})$ .

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Step 1.3.1

Substitute $[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] - λ I_{2})$

Step 1.3.2

Substitute $[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$ for $I_{2}$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] - λ [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}])$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $- λ$ by each element of the matrix.

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ & 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 λ & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

Step 1.4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 6 - λ & 8 + 0 \\ 2 + 0 & 5 - λ \end{array}]$

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Add $8$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 6 - λ & 8 \\ 2 + 0 & 5 - λ \end{array}]$

Step 1.4.3.2

Add $2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 6 - λ & 8 \\ 2 & 5 - λ \end{array}]$

Step 1.5

Find the determinant.

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Step 1.5.1

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

$p (λ) = (6 - λ) (5 - λ) - 2 \cdot 8$

Step 1.5.2

Simplify the determinant.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

Expand $(6 - λ) (5 - λ)$ using the FOIL Method.

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Step 1.5.2.1.1.1

Apply the distributive property.

$p (λ) = 6 (5 - λ) - λ (5 - λ) - 2 \cdot 8$

Step 1.5.2.1.1.2

Apply the distributive property.

$p (λ) = 6 \cdot 5 + 6 (- λ) - λ (5 - λ) - 2 \cdot 8$

Step 1.5.2.1.1.3

Apply the distributive property.

$p (λ) = 6 \cdot 5 + 6 (- λ) - λ \cdot 5 - λ (- λ) - 2 \cdot 8$

Step 1.5.2.1.2

Simplify and combine like terms.

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Step 1.5.2.1.2.1

Simplify each term.

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Step 1.5.2.1.2.1.1

Multiply $6$ by $5$ .

$p (λ) = 30 + 6 (- λ) - λ \cdot 5 - λ (- λ) - 2 \cdot 8$

Step 1.5.2.1.2.1.2

Multiply $- 1$ by $6$ .

$p (λ) = 30 - 6 λ - λ \cdot 5 - λ (- λ) - 2 \cdot 8$

Step 1.5.2.1.2.1.3

Multiply $5$ by $- 1$ .

$p (λ) = 30 - 6 λ - 5 λ - λ (- λ) - 2 \cdot 8$

Step 1.5.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = 30 - 6 λ - 5 λ - 1 \cdot - 1 λ \cdot λ - 2 \cdot 8$

Step 1.5.2.1.2.1.5

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

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Step 1.5.2.1.2.1.5.1

Move $λ$ .

$p (λ) = 30 - 6 λ - 5 λ - 1 \cdot - 1 (λ \cdot λ) - 2 \cdot 8$

Step 1.5.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = 30 - 6 λ - 5 λ - 1 \cdot - 1 λ^{2} - 2 \cdot 8$

Step 1.5.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = 30 - 6 λ - 5 λ + 1 λ^{2} - 2 \cdot 8$

Step 1.5.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = 30 - 6 λ - 5 λ + λ^{2} - 2 \cdot 8$

Step 1.5.2.1.2.2

Subtract $5 λ$ from $- 6 λ$ .

$p (λ) = 30 - 11 λ + λ^{2} - 2 \cdot 8$

Step 1.5.2.1.3

Multiply $- 2$ by $8$ .

$p (λ) = 30 - 11 λ + λ^{2} - 16$

Step 1.5.2.2

Subtract $16$ from $30$ .

$p (λ) = - 11 λ + λ^{2} + 14$

Step 1.5.2.3

Reorder $- 11 λ$ and $λ^{2}$ .

$p (λ) = λ^{2} - 11 λ + 14$

Step 1.6

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

$λ^{2} - 11 λ + 14 = 0$

Step 1.7

Solve for $λ$ .

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Step 1.7.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.7.2

Substitute the values $a = 1$ , $b = - 11$ , and $c = 14$ into the quadratic formula and solve for $λ$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (1 \cdot 14)}}{2 \cdot 1}$

Step 1.7.3

Simplify.

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Step 1.7.3.1

Simplify the numerator.

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Step 1.7.3.1.1

Raise $- 11$ to the power of $2$ .

$λ = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Step 1.7.3.1.2

Multiply $- 4 \cdot 1 \cdot 14$ .

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Step 1.7.3.1.2.1

Multiply $- 4$ by $1$ .

$λ = \frac{11 \pm \sqrt{121 - 4 \cdot 14}}{2 \cdot 1}$

Step 1.7.3.1.2.2

Multiply $- 4$ by $14$ .

$λ = \frac{11 \pm \sqrt{121 - 56}}{2 \cdot 1}$

Step 1.7.3.1.3

Subtract $56$ from $121$ .

$λ = \frac{11 \pm \sqrt{65}}{2 \cdot 1}$

Step 1.7.3.2

Multiply $2$ by $1$ .

$λ = \frac{11 \pm \sqrt{65}}{2}$

Step 1.7.4

The final answer is the combination of both solutions.

$λ = \frac{11 + \sqrt{65}}{2}, \frac{11 - \sqrt{65}}{2}$

Step 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where $N$ is the null space and $I$ is the identity matrix.

$ε_{A} = N (A - λ I_{2})$

Step 3

Find the eigenvector using the eigenvalue $λ = \frac{11 + \sqrt{65}}{2}$ .

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Step 3.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] - \frac{11 + \sqrt{65}}{2} [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}])$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- \frac{11 + \sqrt{65}}{2}$ by each element of the matrix.

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} \cdot 1 & - \frac{11 + \sqrt{65}}{2} \cdot 0 \\ - \frac{11 + \sqrt{65}}{2} \cdot 0 & - \frac{11 + \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply $- 1$ by $1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} & - \frac{11 + \sqrt{65}}{2} \cdot 0 \\ - \frac{11 + \sqrt{65}}{2} \cdot 0 & - \frac{11 + \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply $- \frac{11 + \sqrt{65}}{2} \cdot 0$ .

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Step 3.2.1.2.2.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} & 0 \frac{11 + \sqrt{65}}{2} \\ - \frac{11 + \sqrt{65}}{2} \cdot 0 & - \frac{11 + \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 3.2.1.2.2.2

Multiply $0$ by $\frac{11 + \sqrt{65}}{2}$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} & 0 \\ - \frac{11 + \sqrt{65}}{2} \cdot 0 & - \frac{11 + \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply $- \frac{11 + \sqrt{65}}{2} \cdot 0$ .

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Step 3.2.1.2.3.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} & 0 \\ 0 \frac{11 + \sqrt{65}}{2} & - \frac{11 + \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 3.2.1.2.3.2

Multiply $0$ by $\frac{11 + \sqrt{65}}{2}$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} & 0 \\ 0 & - \frac{11 + \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply $- 1$ by $1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 + \sqrt{65}}{2} & 0 \\ 0 & - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.2

Add the corresponding elements.

$[\begin{array}{c} 6 - \frac{11 + \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3

Simplify each element.

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Step 3.2.3.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$[\begin{array}{c} 6 \cdot \frac{2}{2} - \frac{11 + \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.2

Combine $6$ and $\frac{2}{2}$ .

$[\begin{array}{c} \frac{6 \cdot 2}{2} - \frac{11 + \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.3

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{6 \cdot 2 - (11 + \sqrt{65})}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.4

Simplify the numerator.

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Step 3.2.3.4.1

Multiply $6$ by $2$ .

$[\begin{array}{c} \frac{12 - (11 + \sqrt{65})}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.4.2

Apply the distributive property.

$[\begin{array}{c} \frac{12 - 1 \cdot 11 - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.4.3

Multiply $- 1$ by $11$ .

$[\begin{array}{c} \frac{12 - 11 - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.4.4

Subtract $11$ from $12$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.5

Add $8$ and $0$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 + 0 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.6

Add $2$ and $0$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & 5 - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.7

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & 5 \cdot \frac{2}{2} - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.8

Combine $5$ and $\frac{2}{2}$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{5 \cdot 2}{2} - \frac{11 + \sqrt{65}}{2} \end{array}]$

Step 3.2.3.9

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{5 \cdot 2 - (11 + \sqrt{65})}{2} \end{array}]$

Step 3.2.3.10

Simplify the numerator.

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Step 3.2.3.10.1

Multiply $5$ by $2$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - (11 + \sqrt{65})}{2} \end{array}]$

Step 3.2.3.10.2

Apply the distributive property.

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - 1 \cdot 11 - \sqrt{65}}{2} \end{array}]$

Step 3.2.3.10.3

Multiply $- 1$ by $11$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - 11 - \sqrt{65}}{2} \end{array}]$

Step 3.2.3.10.4

Subtract $11$ from $10$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 - \sqrt{65}}{2} \end{array}]$

Step 3.2.3.11

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

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 (1) - \sqrt{65}}{2} \end{array}]$

Step 3.2.3.12

Factor $- 1$ out of $- \sqrt{65}$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 (1) - (\sqrt{65})}{2} \end{array}]$

Step 3.2.3.13

Factor $- 1$ out of $- 1 (1) - (\sqrt{65})$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 (1 + \sqrt{65})}{2} \end{array}]$

Step 3.2.3.14

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 \\ 2 & - \frac{1 + \sqrt{65}}{2} \end{array}]$

Step 3.3

Find the null space when $λ = \frac{11 + \sqrt{65}}{2}$ .

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Step 3.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} \frac{1 - \sqrt{65}}{2} & 8 & 0 \\ 2 & - \frac{1 + \sqrt{65}}{2} & 0 \end{array}]$

Step 3.3.2

Find the reduced row echelon form.

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Step 3.3.2.1

Multiply each element of $R_{1}$ by $\frac{2}{1 - \sqrt{65}}$ to make the entry at $1, 1$ a $1$ .

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Step 3.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{2}{1 - \sqrt{65}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{2}{1 - \sqrt{65}} \cdot \frac{1 - \sqrt{65}}{2} & \frac{2}{1 - \sqrt{65}} \cdot 8 & \frac{2}{1 - \sqrt{65}} \cdot 0 \\ 2 & - \frac{1 + \sqrt{65}}{2} & 0 \end{array}]$

Step 3.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1 + \sqrt{65}}{4} & 0 \\ 2 & - \frac{1 + \sqrt{65}}{2} & 0 \end{array}]$

Step 3.3.2.2

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.3.2.2.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1 + \sqrt{65}}{4} & 0 \\ 2 - 2 \cdot 1 & - \frac{1 + \sqrt{65}}{2} - 2 (- \frac{1 + \sqrt{65}}{4}) & 0 - 2 \cdot 0 \end{array}]$

Step 3.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1 + \sqrt{65}}{4} & 0 \\ 0 & 0 & 0 \end{array}]$

Step 3.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - \frac{1 + \sqrt{65}}{4} y = 0$

$0 = 0$

Step 3.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{y}{4} + \frac{y \sqrt{65}}{4} \\ y \end{array}]$

Step 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \end{array}] = y [\begin{array}{c} \frac{1}{4} + \frac{\sqrt{65}}{4} \\ 1 \end{array}]$

Step 3.3.6

Write as a solution set.

${y [\begin{array}{c} \frac{1}{4} + \frac{\sqrt{65}}{4} \\ 1 \end{array}] | y \in R}$

Step 3.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} \frac{1}{4} + \frac{\sqrt{65}}{4} \\ 1 \end{array}]}$

Step 4

Find the eigenvector using the eigenvalue $λ = \frac{11 - \sqrt{65}}{2}$ .

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Step 4.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] - \frac{11 - \sqrt{65}}{2} [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}])$

Step 4.2

Simplify.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $- \frac{11 - \sqrt{65}}{2}$ by each element of the matrix.

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} \cdot 1 & - \frac{11 - \sqrt{65}}{2} \cdot 0 \\ - \frac{11 - \sqrt{65}}{2} \cdot 0 & - \frac{11 - \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 4.2.1.2

Simplify each element in the matrix.

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Step 4.2.1.2.1

Multiply $- 1$ by $1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} & - \frac{11 - \sqrt{65}}{2} \cdot 0 \\ - \frac{11 - \sqrt{65}}{2} \cdot 0 & - \frac{11 - \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 4.2.1.2.2

Multiply $- \frac{11 - \sqrt{65}}{2} \cdot 0$ .

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Step 4.2.1.2.2.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} & 0 \frac{11 - \sqrt{65}}{2} \\ - \frac{11 - \sqrt{65}}{2} \cdot 0 & - \frac{11 - \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 4.2.1.2.2.2

Multiply $0$ by $\frac{11 - \sqrt{65}}{2}$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} & 0 \\ - \frac{11 - \sqrt{65}}{2} \cdot 0 & - \frac{11 - \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 4.2.1.2.3

Multiply $- \frac{11 - \sqrt{65}}{2} \cdot 0$ .

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Step 4.2.1.2.3.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} & 0 \\ 0 \frac{11 - \sqrt{65}}{2} & - \frac{11 - \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 4.2.1.2.3.2

Multiply $0$ by $\frac{11 - \sqrt{65}}{2}$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} & 0 \\ 0 & - \frac{11 - \sqrt{65}}{2} \cdot 1 \end{array}]$

Step 4.2.1.2.4

Multiply $- 1$ by $1$ .

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}] + [\begin{array}{c} - \frac{11 - \sqrt{65}}{2} & 0 \\ 0 & - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.2

Add the corresponding elements.

$[\begin{array}{c} 6 - \frac{11 - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3

Simplify each element.

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Step 4.2.3.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$[\begin{array}{c} 6 \cdot \frac{2}{2} - \frac{11 - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.2

Combine $6$ and $\frac{2}{2}$ .

$[\begin{array}{c} \frac{6 \cdot 2}{2} - \frac{11 - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.3

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{6 \cdot 2 - (11 - \sqrt{65})}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.4

Simplify the numerator.

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Step 4.2.3.4.1

Multiply $6$ by $2$ .

$[\begin{array}{c} \frac{12 - (11 - \sqrt{65})}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.4.2

Apply the distributive property.

$[\begin{array}{c} \frac{12 - 1 \cdot 11 - - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.4.3

Multiply $- 1$ by $11$ .

$[\begin{array}{c} \frac{12 - 11 - - \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.4.4

Multiply $- - \sqrt{65}$ .

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Step 4.2.3.4.4.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{12 - 11 + 1 \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.4.4.2

Multiply $\sqrt{65}$ by $1$ .

$[\begin{array}{c} \frac{12 - 11 + \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.4.5

Subtract $11$ from $12$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 + 0 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.5

Add $8$ and $0$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 + 0 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.6

Add $2$ and $0$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & 5 - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.7

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & 5 \cdot \frac{2}{2} - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.8

Combine $5$ and $\frac{2}{2}$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{5 \cdot 2}{2} - \frac{11 - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.9

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{5 \cdot 2 - (11 - \sqrt{65})}{2} \end{array}]$

Step 4.2.3.10

Simplify the numerator.

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Step 4.2.3.10.1

Multiply $5$ by $2$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - (11 - \sqrt{65})}{2} \end{array}]$

Step 4.2.3.10.2

Apply the distributive property.

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - 1 \cdot 11 - - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.10.3

Multiply $- 1$ by $11$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - 11 - - \sqrt{65}}{2} \end{array}]$

Step 4.2.3.10.4

Multiply $- - \sqrt{65}$ .

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Step 4.2.3.10.4.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - 11 + 1 \sqrt{65}}{2} \end{array}]$

Step 4.2.3.10.4.2

Multiply $\sqrt{65}$ by $1$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{10 - 11 + \sqrt{65}}{2} \end{array}]$

Step 4.2.3.10.5

Subtract $11$ from $10$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 + \sqrt{65}}{2} \end{array}]$

Step 4.2.3.11

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

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 (1) + \sqrt{65}}{2} \end{array}]$

Step 4.2.3.12

Factor $- 1$ out of $\sqrt{65}$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 (1) - 1 (- \sqrt{65})}{2} \end{array}]$

Step 4.2.3.13

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{65})$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & \frac{- 1 (1 - \sqrt{65})}{2} \end{array}]$

Step 4.2.3.14

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 \\ 2 & - \frac{1 - \sqrt{65}}{2} \end{array}]$

Step 4.3

Find the null space when $λ = \frac{11 - \sqrt{65}}{2}$ .

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Step 4.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} \frac{1 + \sqrt{65}}{2} & 8 & 0 \\ 2 & - \frac{1 - \sqrt{65}}{2} & 0 \end{array}]$

Step 4.3.2

Find the reduced row echelon form.

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Step 4.3.2.1

Multiply each element of $R_{1}$ by $\frac{2}{1 + \sqrt{65}}$ to make the entry at $1, 1$ a $1$ .

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Step 4.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{2}{1 + \sqrt{65}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{2}{1 + \sqrt{65}} \cdot \frac{1 + \sqrt{65}}{2} & \frac{2}{1 + \sqrt{65}} \cdot 8 & \frac{2}{1 + \sqrt{65}} \cdot 0 \\ 2 & - \frac{1 - \sqrt{65}}{2} & 0 \end{array}]$

Step 4.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1 - \sqrt{65}}{4} & 0 \\ 2 & - \frac{1 - \sqrt{65}}{2} & 0 \end{array}]$

Step 4.3.2.2

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 4.3.2.2.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1 - \sqrt{65}}{4} & 0 \\ 2 - 2 \cdot 1 & - \frac{1 - \sqrt{65}}{2} - 2 (- \frac{1 - \sqrt{65}}{4}) & 0 - 2 \cdot 0 \end{array}]$

Step 4.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1 - \sqrt{65}}{4} & 0 \\ 0 & 0 & 0 \end{array}]$

Step 4.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - \frac{1 - \sqrt{65}}{4} y = 0$

$0 = 0$

Step 4.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{y}{4} - \frac{y \sqrt{65}}{4} \\ y \end{array}]$

Step 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \end{array}] = y [\begin{array}{c} \frac{1}{4} - \frac{\sqrt{65}}{4} \\ 1 \end{array}]$

Step 4.3.6

Write as a solution set.

${y [\begin{array}{c} \frac{1}{4} - \frac{\sqrt{65}}{4} \\ 1 \end{array}] | y \in R}$

Step 4.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} \frac{1}{4} - \frac{\sqrt{65}}{4} \\ 1 \end{array}]}$

Step 5

The eigenspace of $A$ is the list of the vector space for each eigenvalue.

${[\begin{array}{c} \frac{1}{4} + \frac{\sqrt{65}}{4} \\ 1 \end{array}], [\begin{array}{c} \frac{1}{4} - \frac{\sqrt{65}}{4} \\ 1 \end{array}]}$

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