Find the Inverse [[2,4,6],[8,10,12],[14,16,18]]

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \end{array}]$

Step 1

Find the determinant.

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

$2 | \begin{array}{c} 10 & 12 \\ 16 & 18 \end{array} | - 4 | \begin{array}{c} 8 & 12 \\ 14 & 18 \end{array} | + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Evaluate $| \begin{array}{c} 10 & 12 \\ 16 & 18 \end{array} |$ .

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

$2 (10 \cdot 18 - 16 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 14 & 18 \end{array} | + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $10$ by $18$ .

$2 (180 - 16 \cdot 12) - 4 | \begin{array}{c} 8 & 12 \\ 14 & 18 \end{array} | + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Multiply $- 16$ by $12$ .

$2 (180 - 192) - 4 | \begin{array}{c} 8 & 12 \\ 14 & 18 \end{array} | + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Subtract $192$ from $180$ .

$2 \cdot - 12 - 4 | \begin{array}{c} 8 & 12 \\ 14 & 18 \end{array} | + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Evaluate $| \begin{array}{c} 8 & 12 \\ 14 & 18 \end{array} |$ .

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

$2 \cdot - 12 - 4 (8 \cdot 18 - 14 \cdot 12) + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $18$ .

$2 \cdot - 12 - 4 (144 - 14 \cdot 12) + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Multiply $- 14$ by $12$ .

$2 \cdot - 12 - 4 (144 - 168) + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Subtract $168$ from $144$ .

$2 \cdot - 12 - 4 \cdot - 24 + 6 | \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$

Evaluate $| \begin{array}{c} 8 & 10 \\ 14 & 16 \end{array} |$ .

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

$2 \cdot - 12 - 4 \cdot - 24 + 6 (8 \cdot 16 - 14 \cdot 10)$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $16$ .

$2 \cdot - 12 - 4 \cdot - 24 + 6 (128 - 14 \cdot 10)$

Multiply $- 14$ by $10$ .

$2 \cdot - 12 - 4 \cdot - 24 + 6 (128 - 140)$

Subtract $140$ from $128$ .

$2 \cdot - 12 - 4 \cdot - 24 + 6 \cdot - 12$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 12$ .

$- 24 - 4 \cdot - 24 + 6 \cdot - 12$

Multiply $- 4$ by $- 24$ .

$- 24 + 96 + 6 \cdot - 12$

Multiply $6$ by $- 12$ .

$- 24 + 96 - 72$

Add $- 24$ and $96$ .

$72 - 72$

Subtract $72$ from $72$ .

$0$

Step 2

There is no inverse because the determinant is $0$ .

Find the Inverse [[2,4,6],[8,10,12],[14,16,18]]

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