Finite Math Examples

$[\begin{array}{c} 1 & 0 & 2 \\ - 1 & 5 & 3 \\ 0 & 4 & 3 \end{array}]$

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.

$| \begin{array}{c} 5 & 3 \\ 4 & 3 \end{array} |$

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

$a_{11} = 5 \cdot 3 - 4 \cdot 3$

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 $5$ by $3$ .

$a_{11} = 15 - 4 \cdot 3$

Step 2.1.2.2.1.2

Multiply $- 4$ by $3$ .

$a_{11} = 15 - 12$

Step 2.1.2.2.2

Subtract $12$ from $15$ .

$a_{11} = 3$

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.

$| \begin{array}{c} - 1 & 3 \\ 0 & 3 \end{array} |$

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

$a_{12} = - 1 \cdot 3 + 0 \cdot 3$

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 $- 1$ by $3$ .

$a_{12} = - 3 + 0 \cdot 3$

Step 2.2.2.2.1.2

Multiply $0$ by $3$ .

$a_{12} = - 3 + 0$

Step 2.2.2.2.2

Add $- 3$ and $0$ .

$a_{12} = - 3$

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.

$| \begin{array}{c} - 1 & 5 \\ 0 & 4 \end{array} |$

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

$a_{13} = - 1 \cdot 4 + 0 \cdot 5$

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 $- 1$ by $4$ .

$a_{13} = - 4 + 0 \cdot 5$

Step 2.3.2.2.1.2

Multiply $0$ by $5$ .

$a_{13} = - 4 + 0$

Step 2.3.2.2.2

Add $- 4$ and $0$ .

$a_{13} = - 4$

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.

$| \begin{array}{c} 0 & 2 \\ 4 & 3 \end{array} |$

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

$a_{21} = 0 \cdot 3 - 4 \cdot 2$

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 $3$ .

$a_{21} = 0 - 4 \cdot 2$

Step 2.4.2.2.1.2

Multiply $- 4$ by $2$ .

$a_{21} = 0 - 8$

Step 2.4.2.2.2

Subtract $8$ from $0$ .

$a_{21} = - 8$

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.

$| \begin{array}{c} 1 & 2 \\ 0 & 3 \end{array} |$

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

$a_{22} = 1 \cdot 3 + 0 \cdot 2$

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 $3$ by $1$ .

$a_{22} = 3 + 0 \cdot 2$

Step 2.5.2.2.1.2

Multiply $0$ by $2$ .

$a_{22} = 3 + 0$

Step 2.5.2.2.2

Add $3$ and $0$ .

$a_{22} = 3$

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.

$| \begin{array}{c} 0 & 2 \\ 5 & 3 \end{array} |$

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

$a_{31} = 0 \cdot 3 - 5 \cdot 2$

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 $3$ .

$a_{31} = 0 - 5 \cdot 2$

Step 2.7.2.2.1.2

Multiply $- 5$ by $2$ .

$a_{31} = 0 - 10$

Step 2.7.2.2.2

Subtract $10$ from $0$ .

$a_{31} = - 10$

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.

$| \begin{array}{c} 1 & 2 \\ - 1 & 3 \end{array} |$

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

$a_{32} = 1 \cdot 3 - (- 1 \cdot 2)$

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 $3$ by $1$ .

$a_{32} = 3 - (- 1 \cdot 2)$

Step 2.8.2.2.1.2

Multiply $- (- 1 \cdot 2)$ .

Tap for more steps...

Step 2.8.2.2.1.2.1

Multiply $- 1$ by $2$ .

$a_{32} = 3 - - 2$

Step 2.8.2.2.1.2.2

Multiply $- 1$ by $- 2$ .

$a_{32} = 3 + 2$

Step 2.8.2.2.2

Add $3$ and $2$ .

$a_{32} = 5$

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.

$| \begin{array}{c} 1 & 0 \\ - 1 & 5 \end{array} |$

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

$a_{33} = 1 \cdot 5 - - 0$

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 $5$ by $1$ .

$a_{33} = 5 - - 0$

Step 2.9.2.2.1.2

Multiply $- - 0$ .

Tap for more steps...

Step 2.9.2.2.1.2.1

Multiply $- 1$ by $0$ .

$a_{33} = 5 - 0$

Step 2.9.2.2.1.2.2

Multiply $- 1$ by $0$ .

$a_{33} = 5 + 0$

Step 2.9.2.2.2

Add $5$ and $0$ .

$a_{33} = 5$

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.

$[\begin{array}{c} 3 & 3 & - 4 \\ 8 & 3 & - 4 \\ - 10 & - 5 & 5 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 3 & 8 & - 10 \\ 3 & 3 & - 5 \\ - 4 & - 4 & 5 \end{array}]$