Find the Cofactor Matrix [[-1,2,5,0],[-5,0,9,7],[9,12,8,-1],[8,-5,0,6]]

$[\begin{array}{c} - 1 & 2 & 5 & 0 \\ - 5 & 0 & 9 & 7 \\ 9 & 12 & 8 & - 1 \\ 8 & - 5 & 0 & 6 \end{array}]$

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.

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

Consider the determinant with the $1 s t$ row and $1 s t$ column deleted.

$a_{11} = | \begin{array}{c} 0 & 9 & 7 \\ 12 & 8 & - 1 \\ - 5 & 0 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{11} = 0 | \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} | - 9 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} |$ .

$a_{11} = 0 - 9 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Evaluate $| \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} |$ .

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

$a_{11} = 0 - 9 (12 \cdot 6 - (- 5 \cdot - 1)) + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $6$ .

$a_{11} = 0 - 9 (72 - (- 5 \cdot - 1)) + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Multiply $- (- 5 \cdot - 1)$ .

Multiply $- 5$ by $- 1$ .

$a_{11} = 0 - 9 (72 - 1 \cdot 5) + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Multiply $- 1$ by $5$ .

$a_{11} = 0 - 9 (72 - 5) + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Subtract $5$ from $72$ .

$a_{11} = 0 - 9 \cdot 67 + 7 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Evaluate $| \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$ .

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

$a_{11} = 0 - 9 \cdot 67 + 7 (12 \cdot 0 - (- 5 \cdot 8))$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $0$ .

$a_{11} = 0 - 9 \cdot 67 + 7 (0 - (- 5 \cdot 8))$

Multiply $- (- 5 \cdot 8)$ .

Multiply $- 5$ by $8$ .

$a_{11} = 0 - 9 \cdot 67 + 7 (0 - - 40)$

Multiply $- 1$ by $- 40$ .

$a_{11} = 0 - 9 \cdot 67 + 7 (0 + 40)$

Add $0$ and $40$ .

$a_{11} = 0 - 9 \cdot 67 + 7 \cdot 40$

Simplify the determinant.

Simplify each term.

Multiply $- 9$ by $67$ .

$a_{11} = 0 - 603 + 7 \cdot 40$

Multiply $7$ by $40$ .

$a_{11} = 0 - 603 + 280$

Subtract $603$ from $0$ .

$a_{11} = - 603 + 280$

Add $- 603$ and $280$ .

$a_{11} = - 323$

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

Consider the determinant with the $1 s t$ row and $2 n d$ column deleted.

$a_{12} = | \begin{array}{c} - 5 & 9 & 7 \\ 9 & 8 & - 1 \\ 8 & 0 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $2 n d$ column by its cofactor and add.

$a_{12} = - 9 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 8 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} |$ .

$a_{12} = - 9 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 8 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} |$ .

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

$a_{12} = - 9 (9 \cdot 6 - 8 \cdot - 1) + 8 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $6$ .

$a_{12} = - 9 (54 - 8 \cdot - 1) + 8 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Multiply $- 8$ by $- 1$ .

$a_{12} = - 9 (54 + 8) + 8 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Add $54$ and $8$ .

$a_{12} = - 9 \cdot 62 + 8 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} |$ .

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

$a_{12} = - 9 \cdot 62 + 8 (- 5 \cdot 6 - 8 \cdot 7) + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $6$ .

$a_{12} = - 9 \cdot 62 + 8 (- 30 - 8 \cdot 7) + 0$

Multiply $- 8$ by $7$ .

$a_{12} = - 9 \cdot 62 + 8 (- 30 - 56) + 0$

Subtract $56$ from $- 30$ .

$a_{12} = - 9 \cdot 62 + 8 \cdot - 86 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 9$ by $62$ .

$a_{12} = - 558 + 8 \cdot - 86 + 0$

Multiply $8$ by $- 86$ .

$a_{12} = - 558 - 688 + 0$

Subtract $688$ from $- 558$ .

$a_{12} = - 1246 + 0$

Add $- 1246$ and $0$ .

$a_{12} = - 1246$

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

Consider the determinant with the $1 s t$ row and $3 r d$ column deleted.

$a_{13} = | \begin{array}{c} - 5 & 0 & 7 \\ 9 & 12 & - 1 \\ 8 & - 5 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{13} = - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} |$ .

$a_{13} = - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0 + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Evaluate $| \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} |$ .

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

$a_{13} = - 5 (12 \cdot 6 - (- 5 \cdot - 1)) + 0 + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $6$ .

$a_{13} = - 5 (72 - (- 5 \cdot - 1)) + 0 + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $- (- 5 \cdot - 1)$ .

Multiply $- 5$ by $- 1$ .

$a_{13} = - 5 (72 - 1 \cdot 5) + 0 + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $- 1$ by $5$ .

$a_{13} = - 5 (72 - 5) + 0 + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Subtract $5$ from $72$ .

$a_{13} = - 5 \cdot 67 + 0 + 7 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Evaluate $| \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$ .

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

$a_{13} = - 5 \cdot 67 + 0 + 7 (9 \cdot - 5 - 8 \cdot 12)$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 5$ .

$a_{13} = - 5 \cdot 67 + 0 + 7 (- 45 - 8 \cdot 12)$

Multiply $- 8$ by $12$ .

$a_{13} = - 5 \cdot 67 + 0 + 7 (- 45 - 96)$

Subtract $96$ from $- 45$ .

$a_{13} = - 5 \cdot 67 + 0 + 7 \cdot - 141$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $67$ .

$a_{13} = - 335 + 0 + 7 \cdot - 141$

Multiply $7$ by $- 141$ .

$a_{13} = - 335 + 0 - 987$

Add $- 335$ and $0$ .

$a_{13} = - 335 - 987$

Subtract $987$ from $- 335$ .

$a_{13} = - 1322$

Calculate the minor for element $a_{14}$ .

Consider the determinant with the $1 s t$ row and $4 t h$ column deleted.

$a_{14} = | \begin{array}{c} - 5 & 0 & 9 \\ 9 & 12 & 8 \\ 8 & - 5 & 0 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{14} = - 5 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} | + 0 | \begin{array}{c} 9 & 8 \\ 8 & 0 \end{array} | + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 9 & 8 \\ 8 & 0 \end{array} |$ .

$a_{14} = - 5 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} | + 0 + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Evaluate $| \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$ .

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

$a_{14} = - 5 (12 \cdot 0 - (- 5 \cdot 8)) + 0 + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $0$ .

$a_{14} = - 5 (0 - (- 5 \cdot 8)) + 0 + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $- (- 5 \cdot 8)$ .

Multiply $- 5$ by $8$ .

$a_{14} = - 5 (0 - - 40) + 0 + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $- 1$ by $- 40$ .

$a_{14} = - 5 (0 + 40) + 0 + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Add $0$ and $40$ .

$a_{14} = - 5 \cdot 40 + 0 + 9 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Evaluate $| \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$ .

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

$a_{14} = - 5 \cdot 40 + 0 + 9 (9 \cdot - 5 - 8 \cdot 12)$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 5$ .

$a_{14} = - 5 \cdot 40 + 0 + 9 (- 45 - 8 \cdot 12)$

Multiply $- 8$ by $12$ .

$a_{14} = - 5 \cdot 40 + 0 + 9 (- 45 - 96)$

Subtract $96$ from $- 45$ .

$a_{14} = - 5 \cdot 40 + 0 + 9 \cdot - 141$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $40$ .

$a_{14} = - 200 + 0 + 9 \cdot - 141$

Multiply $9$ by $- 141$ .

$a_{14} = - 200 + 0 - 1269$

Add $- 200$ and $0$ .

$a_{14} = - 200 - 1269$

Subtract $1269$ from $- 200$ .

$a_{14} = - 1469$

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

Consider the determinant with the $2 n d$ row and $1 s t$ column deleted.

$a_{21} = | \begin{array}{c} 2 & 5 & 0 \\ 12 & 8 & - 1 \\ - 5 & 0 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{21} = 2 | \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0 | \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 12 & 8 \\ - 5 & 0 \end{array} |$ .

$a_{21} = 2 | \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} |$ .

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

$a_{21} = 2 (8 \cdot 6 + 0 \cdot - 1) - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $6$ .

$a_{21} = 2 (48 + 0 \cdot - 1) - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0$

Multiply $0$ by $- 1$ .

$a_{21} = 2 (48 + 0) - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0$

Add $48$ and $0$ .

$a_{21} = 2 \cdot 48 - 5 | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} |$ .

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

$a_{21} = 2 \cdot 48 - 5 (12 \cdot 6 - (- 5 \cdot - 1)) + 0$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $6$ .

$a_{21} = 2 \cdot 48 - 5 (72 - (- 5 \cdot - 1)) + 0$

Multiply $- (- 5 \cdot - 1)$ .

Multiply $- 5$ by $- 1$ .

$a_{21} = 2 \cdot 48 - 5 (72 - 1 \cdot 5) + 0$

Multiply $- 1$ by $5$ .

$a_{21} = 2 \cdot 48 - 5 (72 - 5) + 0$

Subtract $5$ from $72$ .

$a_{21} = 2 \cdot 48 - 5 \cdot 67 + 0$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $48$ .

$a_{21} = 96 - 5 \cdot 67 + 0$

Multiply $- 5$ by $67$ .

$a_{21} = 96 - 335 + 0$

Subtract $335$ from $96$ .

$a_{21} = - 239 + 0$

Add $- 239$ and $0$ .

$a_{21} = - 239$

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

Consider the determinant with the $2 n d$ row and $2 n d$ column deleted.

$a_{22} = | \begin{array}{c} - 1 & 5 & 0 \\ 9 & 8 & - 1 \\ 8 & 0 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{22} = - | \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0 | \begin{array}{c} 9 & 8 \\ 8 & 0 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 9 & 8 \\ 8 & 0 \end{array} |$ .

$a_{22} = - | \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 8 & - 1 \\ 0 & 6 \end{array} |$ .

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

$a_{22} = - (8 \cdot 6 + 0 \cdot - 1) - 5 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $6$ .

$a_{22} = - (48 + 0 \cdot - 1) - 5 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Multiply $0$ by $- 1$ .

$a_{22} = - (48 + 0) - 5 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Add $48$ and $0$ .

$a_{22} = - 1 \cdot 48 - 5 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} |$ .

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

$a_{22} = - 1 \cdot 48 - 5 (9 \cdot 6 - 8 \cdot - 1) + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $6$ .

$a_{22} = - 1 \cdot 48 - 5 (54 - 8 \cdot - 1) + 0$

Multiply $- 8$ by $- 1$ .

$a_{22} = - 1 \cdot 48 - 5 (54 + 8) + 0$

Add $54$ and $8$ .

$a_{22} = - 1 \cdot 48 - 5 \cdot 62 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $48$ .

$a_{22} = - 48 - 5 \cdot 62 + 0$

Multiply $- 5$ by $62$ .

$a_{22} = - 48 - 310 + 0$

Subtract $310$ from $- 48$ .

$a_{22} = - 358 + 0$

Add $- 358$ and $0$ .

$a_{22} = - 358$

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

Consider the determinant with the $2 n d$ row and $3 r d$ column deleted.

$a_{23} = | \begin{array}{c} - 1 & 2 & 0 \\ 9 & 12 & - 1 \\ 8 & - 5 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{23} = - | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0 | \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 9 & 12 \\ 8 & - 5 \end{array} |$ .

$a_{23} = - | \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} | - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 12 & - 1 \\ - 5 & 6 \end{array} |$ .

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

$a_{23} = - (12 \cdot 6 - (- 5 \cdot - 1)) - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $6$ .

$a_{23} = - (72 - (- 5 \cdot - 1)) - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Multiply $- (- 5 \cdot - 1)$ .

Multiply $- 5$ by $- 1$ .

$a_{23} = - (72 - 1 \cdot 5) - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Multiply $- 1$ by $5$ .

$a_{23} = - (72 - 5) - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Subtract $5$ from $72$ .

$a_{23} = - 1 \cdot 67 - 2 | \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & - 1 \\ 8 & 6 \end{array} |$ .

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

$a_{23} = - 1 \cdot 67 - 2 (9 \cdot 6 - 8 \cdot - 1) + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $6$ .

$a_{23} = - 1 \cdot 67 - 2 (54 - 8 \cdot - 1) + 0$

Multiply $- 8$ by $- 1$ .

$a_{23} = - 1 \cdot 67 - 2 (54 + 8) + 0$

Add $54$ and $8$ .

$a_{23} = - 1 \cdot 67 - 2 \cdot 62 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $67$ .

$a_{23} = - 67 - 2 \cdot 62 + 0$

Multiply $- 2$ by $62$ .

$a_{23} = - 67 - 124 + 0$

Subtract $124$ from $- 67$ .

$a_{23} = - 191 + 0$

Add $- 191$ and $0$ .

$a_{23} = - 191$

Calculate the minor for element $a_{24}$ .

Consider the determinant with the $2 n d$ row and $4 t h$ column deleted.

$a_{24} = | \begin{array}{c} - 1 & 2 & 5 \\ 9 & 12 & 8 \\ 8 & - 5 & 0 \end{array} |$

Evaluate the determinant.

Multiply every element in the $3 r d$ row by its cofactor and add.

$a_{24} = 8 | \begin{array}{c} 2 & 5 \\ 12 & 8 \end{array} | + 5 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | + 0 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$ .

$a_{24} = 8 | \begin{array}{c} 2 & 5 \\ 12 & 8 \end{array} | + 5 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | + 0$

Evaluate $| \begin{array}{c} 2 & 5 \\ 12 & 8 \end{array} |$ .

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

$a_{24} = 8 (2 \cdot 8 - 12 \cdot 5) + 5 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $8$ .

$a_{24} = 8 (16 - 12 \cdot 5) + 5 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | + 0$

Multiply $- 12$ by $5$ .

$a_{24} = 8 (16 - 60) + 5 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | + 0$

Subtract $60$ from $16$ .

$a_{24} = 8 \cdot - 44 + 5 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | + 0$

Evaluate $| \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} |$ .

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

$a_{24} = 8 \cdot - 44 + 5 (- 1 \cdot 8 - 9 \cdot 5) + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $8$ .

$a_{24} = 8 \cdot - 44 + 5 (- 8 - 9 \cdot 5) + 0$

Multiply $- 9$ by $5$ .

$a_{24} = 8 \cdot - 44 + 5 (- 8 - 45) + 0$

Subtract $45$ from $- 8$ .

$a_{24} = 8 \cdot - 44 + 5 \cdot - 53 + 0$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $- 44$ .

$a_{24} = - 352 + 5 \cdot - 53 + 0$

Multiply $5$ by $- 53$ .

$a_{24} = - 352 - 265 + 0$

Subtract $265$ from $- 352$ .

$a_{24} = - 617 + 0$

Add $- 617$ and $0$ .

$a_{24} = - 617$

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

Consider the determinant with the $3 r d$ row and $1 s t$ column deleted.

$a_{31} = | \begin{array}{c} 2 & 5 & 0 \\ 0 & 9 & 7 \\ - 5 & 0 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{31} = 2 | \begin{array}{c} 9 & 7 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | + 0 | \begin{array}{c} 0 & 9 \\ - 5 & 0 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 0 & 9 \\ - 5 & 0 \end{array} |$ .

$a_{31} = 2 | \begin{array}{c} 9 & 7 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & 7 \\ 0 & 6 \end{array} |$ .

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

$a_{31} = 2 (9 \cdot 6 + 0 \cdot 7) - 5 | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $6$ .

$a_{31} = 2 (54 + 0 \cdot 7) - 5 | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | + 0$

Multiply $0$ by $7$ .

$a_{31} = 2 (54 + 0) - 5 | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | + 0$

Add $54$ and $0$ .

$a_{31} = 2 \cdot 54 - 5 | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} |$ .

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

$a_{31} = 2 \cdot 54 - 5 (0 \cdot 6 - (- 5 \cdot 7)) + 0$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $6$ .

$a_{31} = 2 \cdot 54 - 5 (0 - (- 5 \cdot 7)) + 0$

Multiply $- (- 5 \cdot 7)$ .

Multiply $- 5$ by $7$ .

$a_{31} = 2 \cdot 54 - 5 (0 - - 35) + 0$

Multiply $- 1$ by $- 35$ .

$a_{31} = 2 \cdot 54 - 5 (0 + 35) + 0$

Add $0$ and $35$ .

$a_{31} = 2 \cdot 54 - 5 \cdot 35 + 0$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $54$ .

$a_{31} = 108 - 5 \cdot 35 + 0$

Multiply $- 5$ by $35$ .

$a_{31} = 108 - 175 + 0$

Subtract $175$ from $108$ .

$a_{31} = - 67 + 0$

Add $- 67$ and $0$ .

$a_{31} = - 67$

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

Consider the determinant with the $3 r d$ row and $2 n d$ column deleted.

$a_{32} = | \begin{array}{c} - 1 & 5 & 0 \\ - 5 & 9 & 7 \\ 8 & 0 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{32} = - | \begin{array}{c} 9 & 7 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0 | \begin{array}{c} - 5 & 9 \\ 8 & 0 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 5 & 9 \\ 8 & 0 \end{array} |$ .

$a_{32} = - | \begin{array}{c} 9 & 7 \\ 0 & 6 \end{array} | - 5 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & 7 \\ 0 & 6 \end{array} |$ .

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

$a_{32} = - (9 \cdot 6 + 0 \cdot 7) - 5 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $6$ .

$a_{32} = - (54 + 0 \cdot 7) - 5 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Multiply $0$ by $7$ .

$a_{32} = - (54 + 0) - 5 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Add $54$ and $0$ .

$a_{32} = - 1 \cdot 54 - 5 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} |$ .

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

$a_{32} = - 1 \cdot 54 - 5 (- 5 \cdot 6 - 8 \cdot 7) + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $6$ .

$a_{32} = - 1 \cdot 54 - 5 (- 30 - 8 \cdot 7) + 0$

Multiply $- 8$ by $7$ .

$a_{32} = - 1 \cdot 54 - 5 (- 30 - 56) + 0$

Subtract $56$ from $- 30$ .

$a_{32} = - 1 \cdot 54 - 5 \cdot - 86 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $54$ .

$a_{32} = - 54 - 5 \cdot - 86 + 0$

Multiply $- 5$ by $- 86$ .

$a_{32} = - 54 + 430 + 0$

Add $- 54$ and $430$ .

$a_{32} = 376 + 0$

Add $376$ and $0$ .

$a_{32} = 376$

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

Consider the determinant with the $3 r d$ row and $3 r d$ column deleted.

$a_{33} = | \begin{array}{c} - 1 & 2 & 0 \\ - 5 & 0 & 7 \\ 8 & - 5 & 6 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{33} = - | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0 | \begin{array}{c} - 5 & 0 \\ 8 & - 5 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 5 & 0 \\ 8 & - 5 \end{array} |$ .

$a_{33} = - | \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} | - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} 0 & 7 \\ - 5 & 6 \end{array} |$ .

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

$a_{33} = - (0 \cdot 6 - (- 5 \cdot 7)) - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $6$ .

$a_{33} = - (0 - (- 5 \cdot 7)) - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Multiply $- (- 5 \cdot 7)$ .

Multiply $- 5$ by $7$ .

$a_{33} = - (0 - - 35) - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Multiply $- 1$ by $- 35$ .

$a_{33} = - (0 + 35) - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Add $0$ and $35$ .

$a_{33} = - 1 \cdot 35 - 2 | \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} | + 0$

Evaluate $| \begin{array}{c} - 5 & 7 \\ 8 & 6 \end{array} |$ .

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

$a_{33} = - 1 \cdot 35 - 2 (- 5 \cdot 6 - 8 \cdot 7) + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $6$ .

$a_{33} = - 1 \cdot 35 - 2 (- 30 - 8 \cdot 7) + 0$

Multiply $- 8$ by $7$ .

$a_{33} = - 1 \cdot 35 - 2 (- 30 - 56) + 0$

Subtract $56$ from $- 30$ .

$a_{33} = - 1 \cdot 35 - 2 \cdot - 86 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $35$ .

$a_{33} = - 35 - 2 \cdot - 86 + 0$

Multiply $- 2$ by $- 86$ .

$a_{33} = - 35 + 172 + 0$

Add $- 35$ and $172$ .

$a_{33} = 137 + 0$

Add $137$ and $0$ .

$a_{33} = 137$

Calculate the minor for element $a_{34}$ .

Consider the determinant with the $3 r d$ row and $4 t h$ column deleted.

$a_{34} = | \begin{array}{c} - 1 & 2 & 5 \\ - 5 & 0 & 9 \\ 8 & - 5 & 0 \end{array} |$

Evaluate the determinant.

Multiply every element in the $2 n d$ row by its cofactor and add.

$a_{34} = 5 | \begin{array}{c} 2 & 5 \\ - 5 & 0 \end{array} | + 0 | \begin{array}{c} - 1 & 5 \\ 8 & 0 \end{array} | - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 1 & 5 \\ 8 & 0 \end{array} |$ .

$a_{34} = 5 | \begin{array}{c} 2 & 5 \\ - 5 & 0 \end{array} | + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Evaluate $| \begin{array}{c} 2 & 5 \\ - 5 & 0 \end{array} |$ .

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

$a_{34} = 5 (2 \cdot 0 - (- 5 \cdot 5)) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $0$ .

$a_{34} = 5 (0 - (- 5 \cdot 5)) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Multiply $- (- 5 \cdot 5)$ .

Multiply $- 5$ by $5$ .

$a_{34} = 5 (0 - - 25) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Multiply $- 1$ by $- 25$ .

$a_{34} = 5 (0 + 25) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Add $0$ and $25$ .

$a_{34} = 5 \cdot 25 + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$

Evaluate $| \begin{array}{c} - 1 & 2 \\ 8 & - 5 \end{array} |$ .

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

$a_{34} = 5 \cdot 25 + 0 - 9 (- - 5 - 8 \cdot 2)$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $- 5$ .

$a_{34} = 5 \cdot 25 + 0 - 9 (5 - 8 \cdot 2)$

Multiply $- 8$ by $2$ .

$a_{34} = 5 \cdot 25 + 0 - 9 (5 - 16)$

Subtract $16$ from $5$ .

$a_{34} = 5 \cdot 25 + 0 - 9 \cdot - 11$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $25$ .

$a_{34} = 125 + 0 - 9 \cdot - 11$

Multiply $- 9$ by $- 11$ .

$a_{34} = 125 + 0 + 99$

Add $125$ and $0$ .

$a_{34} = 125 + 99$

Add $125$ and $99$ .

$a_{34} = 224$

Calculate the minor for element $a_{41}$ .

Consider the determinant with the $4 t h$ row and $1 s t$ column deleted.

$a_{41} = | \begin{array}{c} 2 & 5 & 0 \\ 0 & 9 & 7 \\ 12 & 8 & - 1 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{41} = 2 | \begin{array}{c} 9 & 7 \\ 8 & - 1 \end{array} | - 5 | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | + 0 | \begin{array}{c} 0 & 9 \\ 12 & 8 \end{array} |$

Multiply $0$ by $| \begin{array}{c} 0 & 9 \\ 12 & 8 \end{array} |$ .

$a_{41} = 2 | \begin{array}{c} 9 & 7 \\ 8 & - 1 \end{array} | - 5 | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & 7 \\ 8 & - 1 \end{array} |$ .

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

$a_{41} = 2 (9 \cdot - 1 - 8 \cdot 7) - 5 | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 1$ .

$a_{41} = 2 (- 9 - 8 \cdot 7) - 5 | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | + 0$

Multiply $- 8$ by $7$ .

$a_{41} = 2 (- 9 - 56) - 5 | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | + 0$

Subtract $56$ from $- 9$ .

$a_{41} = 2 \cdot - 65 - 5 | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | + 0$

Evaluate $| \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} |$ .

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

$a_{41} = 2 \cdot - 65 - 5 (0 \cdot - 1 - 12 \cdot 7) + 0$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $- 1$ .

$a_{41} = 2 \cdot - 65 - 5 (0 - 12 \cdot 7) + 0$

Multiply $- 12$ by $7$ .

$a_{41} = 2 \cdot - 65 - 5 (0 - 84) + 0$

Subtract $84$ from $0$ .

$a_{41} = 2 \cdot - 65 - 5 \cdot - 84 + 0$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 65$ .

$a_{41} = - 130 - 5 \cdot - 84 + 0$

Multiply $- 5$ by $- 84$ .

$a_{41} = - 130 + 420 + 0$

Add $- 130$ and $420$ .

$a_{41} = 290 + 0$

Add $290$ and $0$ .

$a_{41} = 290$

Calculate the minor for element $a_{42}$ .

Consider the determinant with the $4 t h$ row and $2 n d$ column deleted.

$a_{42} = | \begin{array}{c} - 1 & 5 & 0 \\ - 5 & 9 & 7 \\ 9 & 8 & - 1 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{42} = - | \begin{array}{c} 9 & 7 \\ 8 & - 1 \end{array} | - 5 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0 | \begin{array}{c} - 5 & 9 \\ 9 & 8 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 5 & 9 \\ 9 & 8 \end{array} |$ .

$a_{42} = - | \begin{array}{c} 9 & 7 \\ 8 & - 1 \end{array} | - 5 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Evaluate $| \begin{array}{c} 9 & 7 \\ 8 & - 1 \end{array} |$ .

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

$a_{42} = - (9 \cdot - 1 - 8 \cdot 7) - 5 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 1$ .

$a_{42} = - (- 9 - 8 \cdot 7) - 5 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Multiply $- 8$ by $7$ .

$a_{42} = - (- 9 - 56) - 5 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Subtract $56$ from $- 9$ .

$a_{42} = - - 65 - 5 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Evaluate $| \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} |$ .

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

$a_{42} = - - 65 - 5 (- 5 \cdot - 1 - 9 \cdot 7) + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $- 1$ .

$a_{42} = - - 65 - 5 (5 - 9 \cdot 7) + 0$

Multiply $- 9$ by $7$ .

$a_{42} = - - 65 - 5 (5 - 63) + 0$

Subtract $63$ from $5$ .

$a_{42} = - - 65 - 5 \cdot - 58 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $- 65$ .

$a_{42} = 65 - 5 \cdot - 58 + 0$

Multiply $- 5$ by $- 58$ .

$a_{42} = 65 + 290 + 0$

Add $65$ and $290$ .

$a_{42} = 355 + 0$

Add $355$ and $0$ .

$a_{42} = 355$

Calculate the minor for element $a_{43}$ .

Consider the determinant with the $4 t h$ row and $3 r d$ column deleted.

$a_{43} = | \begin{array}{c} - 1 & 2 & 0 \\ - 5 & 0 & 7 \\ 9 & 12 & - 1 \end{array} |$

Evaluate the determinant.

Multiply every element in the $1 s t$ row by its cofactor and add.

$a_{43} = - | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | - 2 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0 | \begin{array}{c} - 5 & 0 \\ 9 & 12 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 5 & 0 \\ 9 & 12 \end{array} |$ .

$a_{43} = - | \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} | - 2 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Evaluate $| \begin{array}{c} 0 & 7 \\ 12 & - 1 \end{array} |$ .

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

$a_{43} = - (0 \cdot - 1 - 12 \cdot 7) - 2 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $- 1$ .

$a_{43} = - (0 - 12 \cdot 7) - 2 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Multiply $- 12$ by $7$ .

$a_{43} = - (0 - 84) - 2 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Subtract $84$ from $0$ .

$a_{43} = - - 84 - 2 | \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} | + 0$

Evaluate $| \begin{array}{c} - 5 & 7 \\ 9 & - 1 \end{array} |$ .

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

$a_{43} = - - 84 - 2 (- 5 \cdot - 1 - 9 \cdot 7) + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $- 1$ .

$a_{43} = - - 84 - 2 (5 - 9 \cdot 7) + 0$

Multiply $- 9$ by $7$ .

$a_{43} = - - 84 - 2 (5 - 63) + 0$

Subtract $63$ from $5$ .

$a_{43} = - - 84 - 2 \cdot - 58 + 0$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $- 84$ .

$a_{43} = 84 - 2 \cdot - 58 + 0$

Multiply $- 2$ by $- 58$ .

$a_{43} = 84 + 116 + 0$

Add $84$ and $116$ .

$a_{43} = 200 + 0$

Add $200$ and $0$ .

$a_{43} = 200$

Calculate the minor for element $a_{44}$ .

Consider the determinant with the $4 t h$ row and $4 t h$ column deleted.

$a_{44} = | \begin{array}{c} - 1 & 2 & 5 \\ - 5 & 0 & 9 \\ 9 & 12 & 8 \end{array} |$

Evaluate the determinant.

Multiply every element in the $2 n d$ row by its cofactor and add.

$a_{44} = 5 | \begin{array}{c} 2 & 5 \\ 12 & 8 \end{array} | + 0 | \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} | - 9 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Multiply $0$ by $| \begin{array}{c} - 1 & 5 \\ 9 & 8 \end{array} |$ .

$a_{44} = 5 | \begin{array}{c} 2 & 5 \\ 12 & 8 \end{array} | + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Evaluate $| \begin{array}{c} 2 & 5 \\ 12 & 8 \end{array} |$ .

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

$a_{44} = 5 (2 \cdot 8 - 12 \cdot 5) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $8$ .

$a_{44} = 5 (16 - 12 \cdot 5) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Multiply $- 12$ by $5$ .

$a_{44} = 5 (16 - 60) + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Subtract $60$ from $16$ .

$a_{44} = 5 \cdot - 44 + 0 - 9 | \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$

Evaluate $| \begin{array}{c} - 1 & 2 \\ 9 & 12 \end{array} |$ .

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

$a_{44} = 5 \cdot - 44 + 0 - 9 (- 1 \cdot 12 - 9 \cdot 2)$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $12$ .

$a_{44} = 5 \cdot - 44 + 0 - 9 (- 12 - 9 \cdot 2)$

Multiply $- 9$ by $2$ .

$a_{44} = 5 \cdot - 44 + 0 - 9 (- 12 - 18)$

Subtract $18$ from $- 12$ .

$a_{44} = 5 \cdot - 44 + 0 - 9 \cdot - 30$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $- 44$ .

$a_{44} = - 220 + 0 - 9 \cdot - 30$

Multiply $- 9$ by $- 30$ .

$a_{44} = - 220 + 0 + 270$

Add $- 220$ and $0$ .

$a_{44} = - 220 + 270$

Add $- 220$ and $270$ .

$a_{44} = 50$

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

$[\begin{array}{c} - 323 & 1246 & - 1322 & 1469 \\ 239 & - 358 & 191 & - 617 \\ - 67 & - 376 & 137 & - 224 \\ - 290 & 355 & - 200 & 50 \end{array}]$

Find the Cofactor Matrix [[-1,2,5,0],[-5,0,9,7],[9,12,8,-1],[8,-5,0,6]]

Related Questions

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