Linear Algebra Examples

$[\begin{array}{c} - 63 & 30 & 24 \\ 33 & - 15 & - 9 \\ - 45 & 30 & 15 \end{array}]$

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.

$| \begin{array}{c} - 15 & - 9 \\ 30 & 15 \end{array} |$

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

$a_{11} = - 15 \cdot 15 - 30 \cdot - 9$

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 $- 15$ by $15$ .

$a_{11} = - 225 - 30 \cdot - 9$

Step 2.1.2.2.1.2

Multiply $- 30$ by $- 9$ .

$a_{11} = - 225 + 270$

Step 2.1.2.2.2

Add $- 225$ and $270$ .

$a_{11} = 45$

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.

$| \begin{array}{c} 33 & - 9 \\ - 45 & 15 \end{array} |$

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

$a_{12} = 33 \cdot 15 - (- 45 \cdot - 9)$

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 $33$ by $15$ .

$a_{12} = 495 - (- 45 \cdot - 9)$

Step 2.2.2.2.1.2

Multiply $- (- 45 \cdot - 9)$ .

Tap for more steps...

Step 2.2.2.2.1.2.1

Multiply $- 45$ by $- 9$ .

$a_{12} = 495 - 1 \cdot 405$

Step 2.2.2.2.1.2.2

Multiply $- 1$ by $405$ .

$a_{12} = 495 - 405$

Step 2.2.2.2.2

Subtract $405$ from $495$ .

$a_{12} = 90$

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.

$| \begin{array}{c} 33 & - 15 \\ - 45 & 30 \end{array} |$

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

$a_{13} = 33 \cdot 30 - (- 45 \cdot - 15)$

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 $33$ by $30$ .

$a_{13} = 990 - (- 45 \cdot - 15)$

Step 2.3.2.2.1.2

Multiply $- (- 45 \cdot - 15)$ .

Tap for more steps...

Step 2.3.2.2.1.2.1

Multiply $- 45$ by $- 15$ .

$a_{13} = 990 - 1 \cdot 675$

Step 2.3.2.2.1.2.2

Multiply $- 1$ by $675$ .

$a_{13} = 990 - 675$

Step 2.3.2.2.2

Subtract $675$ from $990$ .

$a_{13} = 315$

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.

$| \begin{array}{c} 30 & 24 \\ 30 & 15 \end{array} |$

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

$a_{21} = 30 \cdot 15 - 30 \cdot 24$

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 $30$ by $15$ .

$a_{21} = 450 - 30 \cdot 24$

Step 2.4.2.2.1.2

Multiply $- 30$ by $24$ .

$a_{21} = 450 - 720$

Step 2.4.2.2.2

Subtract $720$ from $450$ .

$a_{21} = - 270$

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.

$| \begin{array}{c} - 63 & 24 \\ - 45 & 15 \end{array} |$

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

$a_{22} = - 63 \cdot 15 - (- 45 \cdot 24)$

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 $- 63$ by $15$ .

$a_{22} = - 945 - (- 45 \cdot 24)$

Step 2.5.2.2.1.2

Multiply $- (- 45 \cdot 24)$ .

Tap for more steps...

Step 2.5.2.2.1.2.1

Multiply $- 45$ by $24$ .

$a_{22} = - 945 - - 1080$

Step 2.5.2.2.1.2.2

Multiply $- 1$ by $- 1080$ .

$a_{22} = - 945 + 1080$

Step 2.5.2.2.2

Add $- 945$ and $1080$ .

$a_{22} = 135$

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.

$| \begin{array}{c} - 63 & 30 \\ - 45 & 30 \end{array} |$

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

$a_{23} = - 63 \cdot 30 - (- 45 \cdot 30)$

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 $- 63$ by $30$ .

$a_{23} = - 1890 - (- 45 \cdot 30)$

Step 2.6.2.2.1.2

Multiply $- (- 45 \cdot 30)$ .

Tap for more steps...

Step 2.6.2.2.1.2.1

Multiply $- 45$ by $30$ .

$a_{23} = - 1890 - - 1350$

Step 2.6.2.2.1.2.2

Multiply $- 1$ by $- 1350$ .

$a_{23} = - 1890 + 1350$

Step 2.6.2.2.2

Add $- 1890$ and $1350$ .

$a_{23} = - 540$

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.

$| \begin{array}{c} 30 & 24 \\ - 15 & - 9 \end{array} |$

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

$a_{31} = 30 \cdot - 9 - (- 15 \cdot 24)$

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 $30$ by $- 9$ .

$a_{31} = - 270 - (- 15 \cdot 24)$

Step 2.7.2.2.1.2

Multiply $- (- 15 \cdot 24)$ .

Tap for more steps...

Step 2.7.2.2.1.2.1

Multiply $- 15$ by $24$ .

$a_{31} = - 270 - - 360$

Step 2.7.2.2.1.2.2

Multiply $- 1$ by $- 360$ .

$a_{31} = - 270 + 360$

Step 2.7.2.2.2

Add $- 270$ and $360$ .

$a_{31} = 90$

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.

$| \begin{array}{c} - 63 & 24 \\ 33 & - 9 \end{array} |$

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

$a_{32} = - 63 \cdot - 9 - 33 \cdot 24$

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 $- 63$ by $- 9$ .

$a_{32} = 567 - 33 \cdot 24$

Step 2.8.2.2.1.2

Multiply $- 33$ by $24$ .

$a_{32} = 567 - 792$

Step 2.8.2.2.2

Subtract $792$ from $567$ .

$a_{32} = - 225$

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.

$| \begin{array}{c} - 63 & 30 \\ 33 & - 15 \end{array} |$

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

$a_{33} = - 63 \cdot - 15 - 33 \cdot 30$

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 $- 63$ by $- 15$ .

$a_{33} = 945 - 33 \cdot 30$

Step 2.9.2.2.1.2

Multiply $- 33$ by $30$ .

$a_{33} = 945 - 990$

Step 2.9.2.2.2

Subtract $990$ from $945$ .

$a_{33} = - 45$

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.

$[\begin{array}{c} 45 & - 90 & 315 \\ 270 & 135 & 540 \\ 90 & 225 & - 45 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 45 & 270 & 90 \\ - 90 & 135 & 225 \\ 315 & 540 & - 45 \end{array}]$