Linear Algebra Examples

$[\begin{array}{c} 8 & \sqrt[8]{5} & 9 \\ 8 x & 4 & 0 \\ 6 & 8 & 7 \end{array}]$

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 $2$ 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 $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} \sqrt[8]{5} & 9 \\ 8 & 7 \end{array} |$

Step 1.4

Multiply element $a_{21}$ by its cofactor.

$- 8 x | \begin{array}{c} \sqrt[8]{5} & 9 \\ 8 & 7 \end{array} |$

Step 1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} |$

Step 1.6

Multiply element $a_{22}$ by its cofactor.

$4 | \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} |$

Step 1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 8 & \sqrt[8]{5} \\ 6 & 8 \end{array} |$

Step 1.8

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 8 & \sqrt[8]{5} \\ 6 & 8 \end{array} |$

Step 1.9

Add the terms together.

$- 8 x | \begin{array}{c} \sqrt[8]{5} & 9 \\ 8 & 7 \end{array} | + 4 | \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} | + 0 | \begin{array}{c} 8 & \sqrt[8]{5} \\ 6 & 8 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 8 & \sqrt[8]{5} \\ 6 & 8 \end{array} |$ .

$- 8 x | \begin{array}{c} \sqrt[8]{5} & 9 \\ 8 & 7 \end{array} | + 4 | \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} | + 0$

Step 3

Evaluate $| \begin{array}{c} \sqrt[8]{5} & 9 \\ 8 & 7 \end{array} |$ .

Tap for more steps...

Step 3.1

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

$- 8 x (\sqrt[8]{5} \cdot 7 - 8 \cdot 9) + 4 | \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} | + 0$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Move $7$ to the left of $\sqrt[8]{5}$ .

$- 8 x (7 \cdot \sqrt[8]{5} - 8 \cdot 9) + 4 | \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} | + 0$

Step 3.2.2

Multiply $- 8$ by $9$ .

$- 8 x (7 \sqrt[8]{5} - 72) + 4 | \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} | + 0$

Step 4

Evaluate $| \begin{array}{c} 8 & 9 \\ 6 & 7 \end{array} |$ .

Tap for more steps...

Step 4.1

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

$- 8 x (7 \sqrt[8]{5} - 72) + 4 (8 \cdot 7 - 6 \cdot 9) + 0$

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 $8$ by $7$ .

$- 8 x (7 \sqrt[8]{5} - 72) + 4 (56 - 6 \cdot 9) + 0$

Step 4.2.1.2

Multiply $- 6$ by $9$ .

$- 8 x (7 \sqrt[8]{5} - 72) + 4 (56 - 54) + 0$

Step 4.2.2

Subtract $54$ from $56$ .

$- 8 x (7 \sqrt[8]{5} - 72) + 4 \cdot 2 + 0$

Step 5

Simplify the determinant.

Tap for more steps...

Step 5.1

Add $- 8 x (7 \sqrt[8]{5} - 72) + 4 \cdot 2$ and $0$ .

$- 8 x (7 \sqrt[8]{5} - 72) + 4 \cdot 2$

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Apply the distributive property.

$- 8 x (7 \sqrt[8]{5}) - 8 x \cdot - 72 + 4 \cdot 2$

Step 5.2.2

Multiply $7$ by $- 8$ .

$- 56 x \sqrt[8]{5} - 8 x \cdot - 72 + 4 \cdot 2$

Step 5.2.3

Multiply $- 72$ by $- 8$ .

$- 56 x \sqrt[8]{5} + 576 x + 4 \cdot 2$

Step 5.2.4

Multiply $4$ by $2$ .

$- 56 x \sqrt[8]{5} + 576 x + 8$