Precalculus Examples

$[\begin{array}{c} 6 & - 7 & 3 \\ - 2 & 3 & - 7 \\ - 5 & 5 & - 8 \end{array}]$

Step 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

Tap for more steps...

Step 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.3

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

$| \begin{array}{c} 3 & - 7 \\ 5 & - 8 \end{array} |$

Step 1.4

Multiply element $a_{11}$ by its cofactor.

$6 | \begin{array}{c} 3 & - 7 \\ 5 & - 8 \end{array} |$

Step 1.5

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

$| \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} |$

Step 1.6

Multiply element $a_{12}$ by its cofactor.

$7 | \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} |$

Step 1.7

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

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

Step 1.8

Multiply element $a_{13}$ by its cofactor.

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

Step 1.9

Add the terms together.

$6 | \begin{array}{c} 3 & - 7 \\ 5 & - 8 \end{array} | + 7 | \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} | + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 3 & - 7 \\ 5 & - 8 \end{array} |$ .

Tap for more steps...

Step 2.1

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

$6 (3 \cdot - 8 - 5 \cdot - 7) + 7 | \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} | + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 2.2

Simplify the determinant.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $3$ by $- 8$ .

$6 (- 24 - 5 \cdot - 7) + 7 | \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} | + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 2.2.1.2

Multiply $- 5$ by $- 7$ .

$6 (- 24 + 35) + 7 | \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} | + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 2.2.2

Add $- 24$ and $35$ .

$6 \cdot 11 + 7 | \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} | + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} - 2 & - 7 \\ - 5 & - 8 \end{array} |$ .

Tap for more steps...

Step 3.1

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

$6 \cdot 11 + 7 (- 2 \cdot - 8 - (- 5 \cdot - 7)) + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 3.2

Simplify the determinant.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $- 2$ by $- 8$ .

$6 \cdot 11 + 7 (16 - (- 5 \cdot - 7)) + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 3.2.1.2

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

Tap for more steps...

Step 3.2.1.2.1

Multiply $- 5$ by $- 7$ .

$6 \cdot 11 + 7 (16 - 1 \cdot 35) + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 3.2.1.2.2

Multiply $- 1$ by $35$ .

$6 \cdot 11 + 7 (16 - 35) + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 3.2.2

Subtract $35$ from $16$ .

$6 \cdot 11 + 7 \cdot - 19 + 3 | \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} - 2 & 3 \\ - 5 & 5 \end{array} |$ .

Tap for more steps...

Step 4.1

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

$6 \cdot 11 + 7 \cdot - 19 + 3 (- 2 \cdot 5 - (- 5 \cdot 3))$

Step 4.2

Simplify the determinant.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $- 2$ by $5$ .

$6 \cdot 11 + 7 \cdot - 19 + 3 (- 10 - (- 5 \cdot 3))$

Step 4.2.1.2

Multiply $- (- 5 \cdot 3)$ .

Tap for more steps...

Step 4.2.1.2.1

Multiply $- 5$ by $3$ .

$6 \cdot 11 + 7 \cdot - 19 + 3 (- 10 - - 15)$

Step 4.2.1.2.2

Multiply $- 1$ by $- 15$ .

$6 \cdot 11 + 7 \cdot - 19 + 3 (- 10 + 15)$

Step 4.2.2

Add $- 10$ and $15$ .

$6 \cdot 11 + 7 \cdot - 19 + 3 \cdot 5$

Step 5

Simplify the determinant.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

Multiply $6$ by $11$ .

$66 + 7 \cdot - 19 + 3 \cdot 5$

Step 5.1.2

Multiply $7$ by $- 19$ .

$66 - 133 + 3 \cdot 5$

Step 5.1.3

Multiply $3$ by $5$ .

$66 - 133 + 15$

Step 5.2

Subtract $133$ from $66$ .

$- 67 + 15$

Step 5.3

Add $- 67$ and $15$ .

$- 52$