Find the Cofactor Matrix [[-2,0,1],[1,2,0],[4,2,1]]

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

Step 1

Refer to the corresponding sign matrix below.

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

Step 2

Use the sign matrix and the given matrix, $A$ , to find the cofactor of each element.

Find the determinant of $[\begin{array}{c} 2 & 0 \\ 2 & 1 \end{array}]$ .

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

$A_{11} = (2) (1) - 2 \cdot 0$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $1$ .

$A_{11} = 2 - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$A_{11} = 2 + 0$

Add $2$ and $0$ .

$A_{11} = 2$

Find the determinant of $- [\begin{array}{c} 1 & 0 \\ 4 & 1 \end{array}]$ .

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

$A_{12} = - ((1) (1) - 4 \cdot 0)$

Simplify the determinant.

Simplify each term.

Multiply $1$ by $1$ .

$A_{12} = - (1 - 4 \cdot 0)$

Multiply $- 4$ by $0$ .

$A_{12} = - (1 + 0)$

Simplify the expression.

Add $1$ and $0$ .

$A_{12} = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$A_{12} = - 1$

Find the determinant of $[\begin{array}{c} 1 & 2 \\ 4 & 2 \end{array}]$ .

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

$A_{13} = (1) (2) - 4 \cdot 2$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $1$ .

$A_{13} = 2 - 4 \cdot 2$

Multiply $- 4$ by $2$ .

$A_{13} = 2 - 8$

Subtract $8$ from $2$ .

$A_{13} = - 6$

Find the determinant of $- [\begin{array}{c} 0 & 1 \\ 2 & 1 \end{array}]$ .

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

$A_{21} = - ((0) (1) - 2 \cdot 1)$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $1$ .

$A_{21} = - (0 - 2 \cdot 1)$

Multiply $- 2$ by $1$ .

$A_{21} = - (0 - 2)$

Simplify the expression.

Subtract $2$ from $0$ .

$A_{21} = 2$

Multiply $- 1$ by $- 2$ .

$A_{21} = 2$

Find the determinant of $[\begin{array}{c} - 2 & 1 \\ 4 & 1 \end{array}]$ .

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

$A_{22} = (- 2) (1) - 4 \cdot 1$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $1$ .

$A_{22} = - 2 - 4 \cdot 1$

Multiply $- 4$ by $1$ .

$A_{22} = - 2 - 4$

Subtract $4$ from $- 2$ .

$A_{22} = - 6$

Find the determinant of $- [\begin{array}{c} - 2 & 0 \\ 4 & 2 \end{array}]$ .

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

$A_{23} = - ((- 2) (2) - 4 \cdot 0)$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $2$ .

$A_{23} = - (- 4 - 4 \cdot 0)$

Multiply $- 4$ by $0$ .

$A_{23} = - (- 4 + 0)$

Simplify the expression.

Add $- 4$ and $0$ .

$A_{23} = 4$

Multiply $- 1$ by $- 4$ .

$A_{23} = 4$

Find the determinant of $[\begin{array}{c} 0 & 1 \\ 2 & 0 \end{array}]$ .

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

$A_{31} = (0) (0) - 2 \cdot 1$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $0$ .

$A_{31} = 0 - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$A_{31} = 0 - 2$

Subtract $2$ from $0$ .

$A_{31} = - 2$

Find the determinant of $- [\begin{array}{c} - 2 & 1 \\ 1 & 0 \end{array}]$ .

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

$A_{32} = - ((- 2) (0) - 1 \cdot 1)$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $0$ .

$A_{32} = - (0 - 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$A_{32} = - (0 - 1)$

Simplify the expression.

Subtract $1$ from $0$ .

$A_{32} = 1$

Multiply $- 1$ by $- 1$ .

$A_{32} = 1$

Find the determinant of $[\begin{array}{c} - 2 & 0 \\ 1 & 2 \end{array}]$ .

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

$A_{33} = (- 2) (2) - 1 \cdot 0$

Simplify the determinant.

Multiply $- 2$ by $2$ .

$A_{33} = - 4 - 1 \cdot 0$

Subtract $0$ from $- 4$ .

$A_{33} = - 4$

The cofactor matrix is a matrix with cofactors as the elements.

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

Find the Cofactor Matrix [[-2,0,1],[1,2,0],[4,2,1]]

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