Linear Algebra Examples

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}]$

Step 1

Find the eigenvalues.

Tap for more steps...

Step 1.1

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

$p (λ) = determinant (A - λ I_{3})$

Step 1.2

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 1.3

Substitute the known values into $p (λ) = determinant (A - λ I_{3})$ .

Tap for more steps...

Step 1.3.1

Substitute $[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] - λ I_{3})$

Step 1.3.2

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

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

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

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2

Simplify each element in the matrix.

Tap for more steps...

Step 1.4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 1.4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 1.4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 1.4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.5

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 1.4.1.2.6.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.6.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 1.4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 1.4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.9

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - λ & 0 & 0 \\ 0 & - λ & 0 \\ 0 & 0 & - λ \end{array}])$

Step 1.4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 0 - λ & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 - λ & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3

Simplify each element.

Tap for more steps...

Step 1.4.3.1

Subtract $λ$ from $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 - λ & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3.2

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 & 0 + 0 \\ 0 + 0 & 16 - λ & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3.3

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 & 0 \\ 0 + 0 & 16 - λ & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3.4

Add $0$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 & 0 \\ 0 & 16 - λ & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3.5

Add $16$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 & 0 \\ 0 & 16 - λ & 16 \\ 16 + 0 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3.6

Add $16$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 & 0 \\ 0 & 16 - λ & 16 \\ 16 & 1 + 0 & - 16 - λ \end{array}]$

Step 1.4.3.7

Add $1$ and $0$ .

$p (λ) = determinant [\begin{array}{c} - λ & 0 & 0 \\ 0 & 16 - λ & 16 \\ 16 & 1 & - 16 - λ \end{array}]$

Step 1.5

Find the determinant.

Tap for more steps...

Step 1.5.1

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 1.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 16 - λ & 16 \\ 1 & - 16 - λ \end{array} |$

Step 1.5.1.4

Multiply element $a_{11}$ by its cofactor.

$- λ | \begin{array}{c} 16 - λ & 16 \\ 1 & - 16 - λ \end{array} |$

Step 1.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 0 & 16 \\ 16 & - 16 - λ \end{array} |$

Step 1.5.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 0 & 16 \\ 16 & - 16 - λ \end{array} |$

Step 1.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 0 & 16 - λ \\ 16 & 1 \end{array} |$

Step 1.5.1.8

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

$0 | \begin{array}{c} 0 & 16 - λ \\ 16 & 1 \end{array} |$

Step 1.5.1.9

Add the terms together.

$p (λ) = - λ | \begin{array}{c} 16 - λ & 16 \\ 1 & - 16 - λ \end{array} | + 0 | \begin{array}{c} 0 & 16 \\ 16 & - 16 - λ \end{array} | + 0 | \begin{array}{c} 0 & 16 - λ \\ 16 & 1 \end{array} |$

Step 1.5.2

Multiply $0$ by $| \begin{array}{c} 0 & 16 \\ 16 & - 16 - λ \end{array} |$ .

$p (λ) = - λ | \begin{array}{c} 16 - λ & 16 \\ 1 & - 16 - λ \end{array} | + 0 + 0 | \begin{array}{c} 0 & 16 - λ \\ 16 & 1 \end{array} |$

Step 1.5.3

Multiply $0$ by $| \begin{array}{c} 0 & 16 - λ \\ 16 & 1 \end{array} |$ .

$p (λ) = - λ | \begin{array}{c} 16 - λ & 16 \\ 1 & - 16 - λ \end{array} | + 0 + 0$

Step 1.5.4

Evaluate $| \begin{array}{c} 16 - λ & 16 \\ 1 & - 16 - λ \end{array} |$ .

Tap for more steps...

Step 1.5.4.1

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

$p (λ) = - λ ((16 - λ) (- 16 - λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2

Simplify the determinant.

Tap for more steps...

Step 1.5.4.2.1

Simplify each term.

Tap for more steps...

Step 1.5.4.2.1.1

Expand $(16 - λ) (- 16 - λ)$ using the FOIL Method.

Tap for more steps...

Step 1.5.4.2.1.1.1

Apply the distributive property.

$p (λ) = - λ (16 (- 16 - λ) - λ (- 16 - λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.1.2

Apply the distributive property.

$p (λ) = - λ (16 \cdot - 16 + 16 (- λ) - λ (- 16 - λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.1.3

Apply the distributive property.

$p (λ) = - λ (16 \cdot - 16 + 16 (- λ) - λ \cdot - 16 - λ (- λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 1.5.4.2.1.2.1

Simplify each term.

Tap for more steps...

Step 1.5.4.2.1.2.1.1

Multiply $16$ by $- 16$ .

$p (λ) = - λ (- 256 + 16 (- λ) - λ \cdot - 16 - λ (- λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.2

Multiply $- 1$ by $16$ .

$p (λ) = - λ (- 256 - 16 λ - λ \cdot - 16 - λ (- λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.3

Multiply $- 16$ by $- 1$ .

$p (λ) = - λ (- 256 - 16 λ + 16 λ - λ (- λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = - λ (- 256 - 16 λ + 16 λ - 1 \cdot - 1 λ \cdot λ - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.5

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

Tap for more steps...

Step 1.5.4.2.1.2.1.5.1

Move $λ$ .

$p (λ) = - λ (- 256 - 16 λ + 16 λ - 1 \cdot - 1 (λ \cdot λ) - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = - λ (- 256 - 16 λ + 16 λ - 1 \cdot - 1 λ^{2} - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = - λ (- 256 - 16 λ + 16 λ + 1 λ^{2} - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = - λ (- 256 - 16 λ + 16 λ + λ^{2} - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.2

Add $- 16 λ$ and $16 λ$ .

$p (λ) = - λ (- 256 + 0 + λ^{2} - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.2.3

Add $- 256$ and $0$ .

$p (λ) = - λ (- 256 + λ^{2} - 1 \cdot 16) + 0 + 0$

Step 1.5.4.2.1.3

Multiply $- 1$ by $16$ .

$p (λ) = - λ (- 256 + λ^{2} - 16) + 0 + 0$

Step 1.5.4.2.2

Subtract $16$ from $- 256$ .

$p (λ) = - λ (λ^{2} - 272) + 0 + 0$

Step 1.5.5

Simplify the determinant.

Tap for more steps...

Step 1.5.5.1

Combine the opposite terms in $- λ (λ^{2} - 272) + 0 + 0$ .

Tap for more steps...

Step 1.5.5.1.1

Add $- λ (λ^{2} - 272)$ and $0$ .

$p (λ) = - λ (λ^{2} - 272) + 0$

Step 1.5.5.1.2

Add $- λ (λ^{2} - 272)$ and $0$ .

$p (λ) = - λ (λ^{2} - 272)$

Step 1.5.5.2

Apply the distributive property.

$p (λ) = - λ \cdot λ^{2} - λ \cdot - 272$

Step 1.5.5.3

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

Tap for more steps...

Step 1.5.5.3.1

Move $λ^{2}$ .

$p (λ) = - (λ^{2} λ) - λ \cdot - 272$

Step 1.5.5.3.2

Multiply $λ^{2}$ by $λ$ .

Tap for more steps...

Step 1.5.5.3.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - (λ^{2} λ^{1}) - λ \cdot - 272$

Step 1.5.5.3.2.2

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

$p (λ) = - λ^{2 + 1} - λ \cdot - 272$

Step 1.5.5.3.3

Add $2$ and $1$ .

$p (λ) = - λ^{3} - λ \cdot - 272$

Step 1.5.5.4

Multiply $- 272$ by $- 1$ .

$p (λ) = - λ^{3} + 272 λ$

Step 1.6

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

$- λ^{3} + 272 λ = 0$

Step 1.7

Solve for $λ$ .

Tap for more steps...

Step 1.7.1

Factor $- λ$ out of $- λ^{3} + 272 λ$ .

Tap for more steps...

Step 1.7.1.1

Factor $- λ$ out of $- λ^{3}$ .

$- λ \cdot λ^{2} + 272 λ = 0$

Step 1.7.1.2

Factor $- λ$ out of $272 λ$ .

$- λ \cdot λ^{2} - λ \cdot - 272 = 0$

Step 1.7.1.3

Factor $- λ$ out of $- λ (λ^{2}) - λ (- 272)$ .

$- λ (λ^{2} - 272) = 0$

Step 1.7.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$λ = 0$

$λ^{2} - 272 = 0$

Step 1.7.3

Set $λ$ equal to $0$ .

$λ = 0$

Step 1.7.4

Set $λ^{2} - 272$ equal to $0$ and solve for $λ$ .

Tap for more steps...

Step 1.7.4.1

Set $λ^{2} - 272$ equal to $0$ .

$λ^{2} - 272 = 0$

Step 1.7.4.2

Solve $λ^{2} - 272 = 0$ for $λ$ .

Tap for more steps...

Step 1.7.4.2.1

Add $272$ to both sides of the equation.

$λ^{2} = 272$

Step 1.7.4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$λ = \pm \sqrt{272}$

Step 1.7.4.2.3

Simplify $\pm \sqrt{272}$ .

Tap for more steps...

Step 1.7.4.2.3.1

Rewrite $272$ as $4^{2} \cdot 17$ .

Tap for more steps...

Step 1.7.4.2.3.1.1

Factor $16$ out of $272$ .

$λ = \pm \sqrt{16 (17)}$

Step 1.7.4.2.3.1.2

Rewrite $16$ as $4^{2}$ .

$λ = \pm \sqrt{4^{2} \cdot 17}$

Step 1.7.4.2.3.2

Pull terms out from under the radical.

$λ = \pm 4 \sqrt{17}$

Step 1.7.4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.7.4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$λ = 4 \sqrt{17}$

Step 1.7.4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$λ = - 4 \sqrt{17}$

Step 1.7.4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$λ = 4 \sqrt{17}, - 4 \sqrt{17}$

Step 1.7.5

The final solution is all the values that make $- λ (λ^{2} - 272) = 0$ true.

$λ = 0, 4 \sqrt{17}, - 4 \sqrt{17}$

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.

$ε_{A} = N (A - λ I_{3})$

Step 3

Find the eigenvector using the eigenvalue $λ = 0$ .

Tap for more steps...

Step 3.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + 0 [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $0$ by each element of the matrix.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 \cdot 1 & 0 \cdot 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

Tap for more steps...

Step 3.2.1.2.1

Multiply $0$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 \cdot 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply $0$ by $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply $0$ by $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 \cdot 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply $0$ by $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.5

Multiply $0$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.6

Multiply $0$ by $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 \cdot 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.7

Multiply $0$ by $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.8

Multiply $0$ by $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.9

Multiply $0$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Step 3.2.2

Adding any matrix to a null matrix is the matrix itself.

Tap for more steps...

Step 3.2.2.1

Add the corresponding elements.

$[\begin{array}{c} 0 + 0 & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 + 0 & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2

Simplify each element.

Tap for more steps...

Step 3.2.2.2.1

Add $0$ and $0$ .

$[\begin{array}{c} 0 & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 + 0 & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.2

Add $0$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 + 0 \\ 0 + 0 & 16 + 0 & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.3

Add $0$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 + 0 & 16 + 0 & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.4

Add $0$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 + 0 & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.5

Add $16$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.6

Add $16$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 + 0 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.7

Add $16$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 + 0 & - 16 + 0 \end{array}]$

Step 3.2.2.2.8

Add $1$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 + 0 \end{array}]$

Step 3.2.2.2.9

Add $- 16$ and $0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}]$

Step 3.3

Find the null space when $λ = 0$ .

Tap for more steps...

Step 3.3.1

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

$[\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 16 & 16 & 0 \\ 16 & 1 & - 16 & 0 \end{array}]$

Step 3.3.2

Find the reduced row echelon form.

Tap for more steps...

Step 3.3.2.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 16 & 1 & - 16 & 0 \\ 0 & 16 & 16 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.2

Multiply each element of $R_{1}$ by $\frac{1}{16}$ to make the entry at $1, 1$ a $1$ .

Tap for more steps...

Step 3.3.2.2.1

Multiply each element of $R_{1}$ by $\frac{1}{16}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{16}{16} & \frac{1}{16} & \frac{- 16}{16} & \frac{0}{16} \\ 0 & 16 & 16 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.2.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{16} & - 1 & 0 \\ 0 & 16 & 16 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{16}$ to make the entry at $2, 2$ a $1$ .

Tap for more steps...

Step 3.3.2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{16}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{16} & - 1 & 0 \\ \frac{0}{16} & \frac{16}{16} & \frac{16}{16} & \frac{0}{16} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{16} & - 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.4

Perform the row operation $R_{1} = R_{1} - \frac{1}{16} R_{2}$ to make the entry at $1, 2$ a $0$ .

Tap for more steps...

Step 3.3.2.4.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{16} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{16} \cdot 0 & \frac{1}{16} - \frac{1}{16} \cdot 1 & - 1 - \frac{1}{16} \cdot 1 & 0 - \frac{1}{16} \cdot 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{17}{16} & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3.3.3

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

$x - \frac{17}{16} z = 0$

$y + z = 0$

$0 = 0$

Step 3.3.4

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{17 z}{16} \\ - z \\ z \end{array}]$

Step 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} \frac{17}{16} \\ - 1 \\ 1 \end{array}]$

Step 3.3.6

Write as a solution set.

${z [\begin{array}{c} \frac{17}{16} \\ - 1 \\ 1 \end{array}] | z \in R}$

Step 3.3.7

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

${[\begin{array}{c} \frac{17}{16} \\ - 1 \\ 1 \end{array}]}$

Step 4

Find the eigenvector using the eigenvalue $λ = 4 \sqrt{17}$ .

Tap for more steps...

Step 4.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] - 4 \sqrt{17} [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $- 4 \sqrt{17}$ by each element of the matrix.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2

Simplify each element in the matrix.

Tap for more steps...

Step 4.2.1.2.1

Multiply $- 4$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.2

Multiply $- 4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.2.1

Multiply $0$ by $- 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 \sqrt{17} & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.2.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.3

Multiply $- 4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.3.1

Multiply $0$ by $- 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \sqrt{17} \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.3.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.4

Multiply $- 4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.4.1

Multiply $0$ by $- 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 \sqrt{17} & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.4.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} \cdot 1 & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.5

Multiply $- 4$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & - 4 \sqrt{17} \cdot 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.6

Multiply $- 4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.6.1

Multiply $0$ by $- 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \sqrt{17} \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.6.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \\ - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.7

Multiply $- 4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.7.1

Multiply $0$ by $- 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \\ 0 \sqrt{17} & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.7.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \\ 0 & - 4 \sqrt{17} \cdot 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.8

Multiply $- 4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.8.1

Multiply $0$ by $- 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \\ 0 & 0 \sqrt{17} & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.8.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \\ 0 & 0 & - 4 \sqrt{17} \cdot 1 \end{array}]$

Step 4.2.1.2.9

Multiply $- 4$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & - 4 \sqrt{17} & 0 \\ 0 & 0 & - 4 \sqrt{17} \end{array}]$

Step 4.2.2

Add the corresponding elements.

$[\begin{array}{c} 0 - 4 \sqrt{17} & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 - 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3

Simplify each element.

Tap for more steps...

Step 4.2.3.1

Subtract $4 \sqrt{17}$ from $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 - 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3.2

Add $0$ and $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 + 0 \\ 0 + 0 & 16 - 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3.3

Add $0$ and $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 + 0 & 16 - 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3.4

Add $0$ and $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3.5

Add $16$ and $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 \\ 16 + 0 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3.6

Add $16$ and $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 \\ 16 & 1 + 0 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.2.3.7

Add $1$ and $0$ .

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 \\ 16 & 1 & - 16 - 4 \sqrt{17} \end{array}]$

Step 4.3

Find the null space when $λ = 4 \sqrt{17}$ .

Tap for more steps...

Step 4.3.1

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

$[\begin{array}{c} - 4 \sqrt{17} & 0 & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 & 0 \\ 16 & 1 & - 16 - 4 \sqrt{17} & 0 \end{array}]$

Step 4.3.2

Find the reduced row echelon form.

Tap for more steps...

Step 4.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{- 4 \sqrt{17}}$ to make the entry at $1, 1$ a $1$ .

Tap for more steps...

Step 4.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{- 4 \sqrt{17}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{- 4 \sqrt{17}}{- 4 \sqrt{17}} & \frac{0}{- 4 \sqrt{17}} & \frac{0}{- 4 \sqrt{17}} & \frac{0}{- 4 \sqrt{17}} \\ 0 & 16 - 4 \sqrt{17} & 16 & 0 \\ 16 & 1 & - 16 - 4 \sqrt{17} & 0 \end{array}]$

Step 4.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 & 0 \\ 16 & 1 & - 16 - 4 \sqrt{17} & 0 \end{array}]$

Step 4.3.2.2

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

Tap for more steps...

Step 4.3.2.2.1

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 & 0 \\ 16 - 16 \cdot 1 & 1 - 16 \cdot 0 & - 16 - 4 \sqrt{17} - 16 \cdot 0 & 0 - 16 \cdot 0 \end{array}]$

Step 4.3.2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 16 - 4 \sqrt{17} & 16 & 0 \\ 0 & 1 & - 16 - 4 \sqrt{17} & 0 \end{array}]$

Step 4.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{16 - 4 \sqrt{17}}$ to make the entry at $2, 2$ a $1$ .

Tap for more steps...

Step 4.3.2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{16 - 4 \sqrt{17}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ \frac{0}{16 - 4 \sqrt{17}} & \frac{16 - 4 \sqrt{17}}{16 - 4 \sqrt{17}} & \frac{16}{16 - 4 \sqrt{17}} & \frac{0}{16 - 4 \sqrt{17}} \\ 0 & 1 & - 16 - 4 \sqrt{17} & 0 \end{array}]$

Step 4.3.2.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - 16 - 4 \sqrt{17} & 0 \\ 0 & 1 & - 16 - 4 \sqrt{17} & 0 \end{array}]$

Step 4.3.2.4

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

Tap for more steps...

Step 4.3.2.4.1

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - 16 - 4 \sqrt{17} & 0 \\ 0 - 0 & 1 - 1 & - 16 - 4 \sqrt{17} - (- 16 - 4 \sqrt{17}) & 0 - 0 \end{array}]$

Step 4.3.2.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - 16 - 4 \sqrt{17} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.3.3

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

$x = 0$

$y + (- 16 - 4 \sqrt{17}) z = 0$

$0 = 0$

Step 4.3.4

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 16 z + 4 \sqrt{17} z \\ z \end{array}]$

Step 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 0 \\ 16 + 4 \sqrt{17} \\ 1 \end{array}]$

Step 4.3.6

Write as a solution set.

${z [\begin{array}{c} 0 \\ 16 + 4 \sqrt{17} \\ 1 \end{array}] | z \in R}$

Step 4.3.7

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

${[\begin{array}{c} 0 \\ 16 + 4 \sqrt{17} \\ 1 \end{array}]}$

Step 5

Find the eigenvector using the eigenvalue $λ = - 4 \sqrt{17}$ .

Tap for more steps...

Step 5.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + 4 \sqrt{17} [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}])$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $4 \sqrt{17}$ by each element of the matrix.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2

Simplify each element in the matrix.

Tap for more steps...

Step 5.2.1.2.1

Multiply $4$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.2

Multiply $4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 5.2.1.2.2.1

Multiply $0$ by $4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 \sqrt{17} & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.2.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.3

Multiply $4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 5.2.1.2.3.1

Multiply $0$ by $4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \sqrt{17} \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.3.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.4

Multiply $4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 5.2.1.2.4.1

Multiply $0$ by $4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 \sqrt{17} & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.4.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} \cdot 1 & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.5

Multiply $4$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 4 \sqrt{17} \cdot 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.6

Multiply $4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 5.2.1.2.6.1

Multiply $0$ by $4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \sqrt{17} \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.6.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \\ 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.7

Multiply $4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 5.2.1.2.7.1

Multiply $0$ by $4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \\ 0 \sqrt{17} & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.7.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \\ 0 & 4 \sqrt{17} \cdot 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.8

Multiply $4 \sqrt{17} \cdot 0$ .

Tap for more steps...

Step 5.2.1.2.8.1

Multiply $0$ by $4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \\ 0 & 0 \sqrt{17} & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.8.2

Multiply $0$ by $\sqrt{17}$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \\ 0 & 0 & 4 \sqrt{17} \cdot 1 \end{array}]$

Step 5.2.1.2.9

Multiply $4$ by $1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 16 & 16 \\ 16 & 1 & - 16 \end{array}] + [\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 4 \sqrt{17} & 0 \\ 0 & 0 & 4 \sqrt{17} \end{array}]$

Step 5.2.2

Add the corresponding elements.

$[\begin{array}{c} 0 + 4 \sqrt{17} & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 + 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3

Simplify each element.

Tap for more steps...

Step 5.2.3.1

Add $0$ and $4 \sqrt{17}$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 + 0 & 0 + 0 \\ 0 + 0 & 16 + 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3.2

Add $0$ and $0$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 + 0 \\ 0 + 0 & 16 + 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3.3

Add $0$ and $0$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 + 0 & 16 + 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3.4

Add $0$ and $0$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 + 0 \\ 16 + 0 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3.5

Add $16$ and $0$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 \\ 16 + 0 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3.6

Add $16$ and $0$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 \\ 16 & 1 + 0 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.2.3.7

Add $1$ and $0$ .

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 \\ 16 & 1 & - 16 + 4 \sqrt{17} \end{array}]$

Step 5.3

Find the null space when $λ = - 4 \sqrt{17}$ .

Tap for more steps...

Step 5.3.1

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

$[\begin{array}{c} 4 \sqrt{17} & 0 & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 & 0 \\ 16 & 1 & - 16 + 4 \sqrt{17} & 0 \end{array}]$

Step 5.3.2

Find the reduced row echelon form.

Tap for more steps...

Step 5.3.2.1

Multiply each element of $R_{1}$ by $\frac{1}{4 \sqrt{17}}$ to make the entry at $1, 1$ a $1$ .

Tap for more steps...

Step 5.3.2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{4 \sqrt{17}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{4 \sqrt{17}}{4 \sqrt{17}} & \frac{0}{4 \sqrt{17}} & \frac{0}{4 \sqrt{17}} & \frac{0}{4 \sqrt{17}} \\ 0 & 16 + 4 \sqrt{17} & 16 & 0 \\ 16 & 1 & - 16 + 4 \sqrt{17} & 0 \end{array}]$

Step 5.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 & 0 \\ 16 & 1 & - 16 + 4 \sqrt{17} & 0 \end{array}]$

Step 5.3.2.2

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

Tap for more steps...

Step 5.3.2.2.1

Perform the row operation $R_{3} = R_{3} - 16 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 & 0 \\ 16 - 16 \cdot 1 & 1 - 16 \cdot 0 & - 16 + 4 \sqrt{17} - 16 \cdot 0 & 0 - 16 \cdot 0 \end{array}]$

Step 5.3.2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 16 + 4 \sqrt{17} & 16 & 0 \\ 0 & 1 & - 16 + 4 \sqrt{17} & 0 \end{array}]$

Step 5.3.2.3

Multiply each element of $R_{2}$ by $\frac{1}{16 + 4 \sqrt{17}}$ to make the entry at $2, 2$ a $1$ .

Tap for more steps...

Step 5.3.2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{16 + 4 \sqrt{17}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ \frac{0}{16 + 4 \sqrt{17}} & \frac{16 + 4 \sqrt{17}}{16 + 4 \sqrt{17}} & \frac{16}{16 + 4 \sqrt{17}} & \frac{0}{16 + 4 \sqrt{17}} \\ 0 & 1 & - 16 + 4 \sqrt{17} & 0 \end{array}]$

Step 5.3.2.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - 16 + 4 \sqrt{17} & 0 \\ 0 & 1 & - 16 + 4 \sqrt{17} & 0 \end{array}]$

Step 5.3.2.4

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

Tap for more steps...

Step 5.3.2.4.1

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - 16 + 4 \sqrt{17} & 0 \\ 0 - 0 & 1 - 1 & - 16 + 4 \sqrt{17} - (- 16 + 4 \sqrt{17}) & 0 - 0 \end{array}]$

Step 5.3.2.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & - 16 + 4 \sqrt{17} & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 5.3.3

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

$x = 0$

$y + (- 16 + 4 \sqrt{17}) z = 0$

$0 = 0$

Step 5.3.4

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 16 z - 4 \sqrt{17} z \\ z \end{array}]$

Step 5.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 0 \\ 16 - 4 \sqrt{17} \\ 1 \end{array}]$

Step 5.3.6

Write as a solution set.

${z [\begin{array}{c} 0 \\ 16 - 4 \sqrt{17} \\ 1 \end{array}] | z \in R}$

Step 5.3.7

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

${[\begin{array}{c} 0 \\ 16 - 4 \sqrt{17} \\ 1 \end{array}]}$

Step 6

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

${[\begin{array}{c} \frac{17}{16} \\ - 1 \\ 1 \end{array}], [\begin{array}{c} 0 \\ 16 + 4 \sqrt{17} \\ 1 \end{array}], [\begin{array}{c} 0 \\ 16 - 4 \sqrt{17} \\ 1 \end{array}]}$