Linear Algebra Examples

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 2 & 6 \\ 0 & - 4 & - 12 \end{array}]$

Step 1

Consider the corresponding sign chart.

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

Step 2

Use the sign chart and the given matrix to find the cofactor of each element.

Tap for more steps...

Step 2.1

Calculate the minor for element $a_{11}$ .

Tap for more steps...

Step 2.1.1

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

$| \begin{array}{c} 2 & 6 \\ - 4 & - 12 \end{array} |$

Step 2.1.2

Evaluate the determinant.

Tap for more steps...

Step 2.1.2.1

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

$a_{11} = 2 \cdot - 12 - (- 4 \cdot 6)$

Step 2.1.2.2

Simplify the determinant.

Tap for more steps...

Step 2.1.2.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.2.1.1

Multiply $2$ by $- 12$ .

$a_{11} = - 24 - (- 4 \cdot 6)$

Step 2.1.2.2.1.2

Multiply $- (- 4 \cdot 6)$ .

Tap for more steps...

Step 2.1.2.2.1.2.1

Multiply $- 4$ by $6$ .

$a_{11} = - 24 - - 24$

Step 2.1.2.2.1.2.2

Multiply $- 1$ by $- 24$ .

$a_{11} = - 24 + 24$

Step 2.1.2.2.2

Add $- 24$ and $24$ .

$a_{11} = 0$

Step 2.2

Calculate the minor for element $a_{12}$ .

Tap for more steps...

Step 2.2.1

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

$| \begin{array}{c} 0 & 6 \\ 0 & - 12 \end{array} |$

Step 2.2.2

Evaluate the determinant.

Tap for more steps...

Step 2.2.2.1

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

$a_{12} = 0 \cdot - 12 + 0 \cdot 6$

Step 2.2.2.2

Simplify the determinant.

Tap for more steps...

Step 2.2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.2.1.1

Multiply $0$ by $- 12$ .

$a_{12} = 0 + 0 \cdot 6$

Step 2.2.2.2.1.2

Multiply $0$ by $6$ .

$a_{12} = 0 + 0$

Step 2.2.2.2.2

Add $0$ and $0$ .

$a_{12} = 0$

Step 2.3

Calculate the minor for element $a_{13}$ .

Tap for more steps...

Step 2.3.1

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

$| \begin{array}{c} 0 & 2 \\ 0 & - 4 \end{array} |$

Step 2.3.2

Evaluate the determinant.

Tap for more steps...

Step 2.3.2.1

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

$a_{13} = 0 \cdot - 4 + 0 \cdot 2$

Step 2.3.2.2

Simplify the determinant.

Tap for more steps...

Step 2.3.2.2.1

Simplify each term.

Tap for more steps...

Step 2.3.2.2.1.1

Multiply $0$ by $- 4$ .

$a_{13} = 0 + 0 \cdot 2$

Step 2.3.2.2.1.2

Multiply $0$ by $2$ .

$a_{13} = 0 + 0$

Step 2.3.2.2.2

Add $0$ and $0$ .

$a_{13} = 0$

Step 2.4

Calculate the minor for element $a_{21}$ .

Tap for more steps...

Step 2.4.1

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 0 & 0 \\ - 4 & - 12 \end{array} |$

Step 2.4.2

Evaluate the determinant.

Tap for more steps...

Step 2.4.2.1

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

$a_{21} = 0 \cdot - 12 - (- 4 \cdot 0)$

Step 2.4.2.2

Simplify the determinant.

Tap for more steps...

Step 2.4.2.2.1

Simplify each term.

Tap for more steps...

Step 2.4.2.2.1.1

Multiply $0$ by $- 12$ .

$a_{21} = 0 - (- 4 \cdot 0)$

Step 2.4.2.2.1.2

Multiply $- (- 4 \cdot 0)$ .

Tap for more steps...

Step 2.4.2.2.1.2.1

Multiply $- 4$ by $0$ .

$a_{21} = 0 - 0$

Step 2.4.2.2.1.2.2

Multiply $- 1$ by $0$ .

$a_{21} = 0 + 0$

Step 2.4.2.2.2

Add $0$ and $0$ .

$a_{21} = 0$

Step 2.5

Calculate the minor for element $a_{22}$ .

Tap for more steps...

Step 2.5.1

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & 0 \\ 0 & - 12 \end{array} |$

Step 2.5.2

Evaluate the determinant.

Tap for more steps...

Step 2.5.2.1

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

$a_{22} = 1 \cdot - 12 + 0 \cdot 0$

Step 2.5.2.2

Simplify the determinant.

Tap for more steps...

Step 2.5.2.2.1

Simplify each term.

Tap for more steps...

Step 2.5.2.2.1.1

Multiply $- 12$ by $1$ .

$a_{22} = - 12 + 0 \cdot 0$

Step 2.5.2.2.1.2

Multiply $0$ by $0$ .

$a_{22} = - 12 + 0$

Step 2.5.2.2.2

Add $- 12$ and $0$ .

$a_{22} = - 12$

Step 2.6

Calculate the minor for element $a_{23}$ .

Tap for more steps...

Step 2.6.1

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 & 0 \\ 0 & - 4 \end{array} |$

Step 2.6.2

Evaluate the determinant.

Tap for more steps...

Step 2.6.2.1

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

$a_{23} = 1 \cdot - 4 + 0 \cdot 0$

Step 2.6.2.2

Simplify the determinant.

Tap for more steps...

Step 2.6.2.2.1

Simplify each term.

Tap for more steps...

Step 2.6.2.2.1.1

Multiply $- 4$ by $1$ .

$a_{23} = - 4 + 0 \cdot 0$

Step 2.6.2.2.1.2

Multiply $0$ by $0$ .

$a_{23} = - 4 + 0$

Step 2.6.2.2.2

Add $- 4$ and $0$ .

$a_{23} = - 4$

Step 2.7

Calculate the minor for element $a_{31}$ .

Tap for more steps...

Step 2.7.1

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 0 & 0 \\ 2 & 6 \end{array} |$

Step 2.7.2

Evaluate the determinant.

Tap for more steps...

Step 2.7.2.1

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

$a_{31} = 0 \cdot 6 - 2 \cdot 0$

Step 2.7.2.2

Simplify the determinant.

Tap for more steps...

Step 2.7.2.2.1

Simplify each term.

Tap for more steps...

Step 2.7.2.2.1.1

Multiply $0$ by $6$ .

$a_{31} = 0 - 2 \cdot 0$

Step 2.7.2.2.1.2

Multiply $- 2$ by $0$ .

$a_{31} = 0 + 0$

Step 2.7.2.2.2

Add $0$ and $0$ .

$a_{31} = 0$

Step 2.8

Calculate the minor for element $a_{32}$ .

Tap for more steps...

Step 2.8.1

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 1 & 0 \\ 0 & 6 \end{array} |$

Step 2.8.2

Evaluate the determinant.

Tap for more steps...

Step 2.8.2.1

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

$a_{32} = 1 \cdot 6 + 0 \cdot 0$

Step 2.8.2.2

Simplify the determinant.

Tap for more steps...

Step 2.8.2.2.1

Simplify each term.

Tap for more steps...

Step 2.8.2.2.1.1

Multiply $6$ by $1$ .

$a_{32} = 6 + 0 \cdot 0$

Step 2.8.2.2.1.2

Multiply $0$ by $0$ .

$a_{32} = 6 + 0$

Step 2.8.2.2.2

Add $6$ and $0$ .

$a_{32} = 6$

Step 2.9

Calculate the minor for element $a_{33}$ .

Tap for more steps...

Step 2.9.1

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

$| \begin{array}{c} 1 & 0 \\ 0 & 2 \end{array} |$

Step 2.9.2

Evaluate the determinant.

Tap for more steps...

Step 2.9.2.1

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

$a_{33} = 1 \cdot 2 + 0 \cdot 0$

Step 2.9.2.2

Simplify the determinant.

Tap for more steps...

Step 2.9.2.2.1

Simplify each term.

Tap for more steps...

Step 2.9.2.2.1.1

Multiply $2$ by $1$ .

$a_{33} = 2 + 0 \cdot 0$

Step 2.9.2.2.1.2

Multiply $0$ by $0$ .

$a_{33} = 2 + 0$

Step 2.9.2.2.2

Add $2$ and $0$ .

$a_{33} = 2$

Step 2.10

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the $-$ positions on the sign chart.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 12 & 4 \\ 0 & - 6 & 2 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 12 & - 6 \\ 0 & 4 & 2 \end{array}]$