Find the Inverse [[-5,6,7],[7,9,8],[2,-3,4]]

$[\begin{array}{c} - 5 & 6 & 7 \\ 7 & 9 & 8 \\ 2 & - 3 & 4 \end{array}]$

Step 1

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} - 5 & 6 & 7 & 1 & 0 & 0 \\ 7 & 9 & 8 & 0 & 1 & 0 \\ 2 & - 3 & 4 & 0 & 0 & 1 \end{array}]$

Step 2

Find the reduced row echelon form.

Multiply each element of $R_{1}$ by $- \frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot 6 & - \frac{1}{5} \cdot 7 & - \frac{1}{5} \cdot 1 & - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot 0 \\ 7 & 9 & 8 & 0 & 1 & 0 \\ 2 & - 3 & 4 & 0 & 0 & 1 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 7 & 9 & 8 & 0 & 1 & 0 \\ 2 & - 3 & 4 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 7 - 7 \cdot 1 & 9 - 7 (- \frac{6}{5}) & 8 - 7 (- \frac{7}{5}) & 0 - 7 (- \frac{1}{5}) & 1 - 7 \cdot 0 & 0 - 7 \cdot 0 \\ 2 & - 3 & 4 & 0 & 0 & 1 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & \frac{87}{5} & \frac{89}{5} & \frac{7}{5} & 1 & 0 \\ 2 & - 3 & 4 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & \frac{87}{5} & \frac{89}{5} & \frac{7}{5} & 1 & 0 \\ 2 - 2 \cdot 1 & - 3 - 2 (- \frac{6}{5}) & 4 - 2 (- \frac{7}{5}) & 0 - 2 (- \frac{1}{5}) & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & \frac{87}{5} & \frac{89}{5} & \frac{7}{5} & 1 & 0 \\ 0 & - \frac{3}{5} & \frac{34}{5} & \frac{2}{5} & 0 & 1 \end{array}]$

Multiply each element of $R_{2}$ by $\frac{5}{87}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ \frac{5}{87} \cdot 0 & \frac{5}{87} \cdot \frac{87}{5} & \frac{5}{87} \cdot \frac{89}{5} & \frac{5}{87} \cdot \frac{7}{5} & \frac{5}{87} \cdot 1 & \frac{5}{87} \cdot 0 \\ 0 & - \frac{3}{5} & \frac{34}{5} & \frac{2}{5} & 0 & 1 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & 1 & \frac{89}{87} & \frac{7}{87} & \frac{5}{87} & 0 \\ 0 & - \frac{3}{5} & \frac{34}{5} & \frac{2}{5} & 0 & 1 \end{array}]$

Perform the row operation $R_{3} = R_{3} + \frac{3}{5} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & 1 & \frac{89}{87} & \frac{7}{87} & \frac{5}{87} & 0 \\ 0 + \frac{3}{5} \cdot 0 & - \frac{3}{5} + \frac{3}{5} \cdot 1 & \frac{34}{5} + \frac{3}{5} \cdot \frac{89}{87} & \frac{2}{5} + \frac{3}{5} \cdot \frac{7}{87} & 0 + \frac{3}{5} \cdot \frac{5}{87} & 1 + \frac{3}{5} \cdot 0 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & 1 & \frac{89}{87} & \frac{7}{87} & \frac{5}{87} & 0 \\ 0 & 0 & \frac{215}{29} & \frac{13}{29} & \frac{1}{29} & 1 \end{array}]$

Multiply each element of $R_{3}$ by $\frac{29}{215}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & 1 & \frac{89}{87} & \frac{7}{87} & \frac{5}{87} & 0 \\ \frac{29}{215} \cdot 0 & \frac{29}{215} \cdot 0 & \frac{29}{215} \cdot \frac{215}{29} & \frac{29}{215} \cdot \frac{13}{29} & \frac{29}{215} \cdot \frac{1}{29} & \frac{29}{215} \cdot 1 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & 1 & \frac{89}{87} & \frac{7}{87} & \frac{5}{87} & 0 \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Perform the row operation $R_{2} = R_{2} - \frac{89}{87} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 - \frac{89}{87} \cdot 0 & 1 - \frac{89}{87} \cdot 0 & \frac{89}{87} - \frac{89}{87} \cdot 1 & \frac{7}{87} - \frac{89}{87} \cdot \frac{13}{215} & \frac{5}{87} - \frac{89}{87} \cdot \frac{1}{215} & 0 - \frac{89}{87} \cdot \frac{29}{215} \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & - \frac{7}{5} & - \frac{1}{5} & 0 & 0 \\ 0 & 1 & 0 & \frac{4}{215} & \frac{34}{645} & - \frac{89}{645} \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Perform the row operation $R_{1} = R_{1} + \frac{7}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + \frac{7}{5} \cdot 0 & - \frac{6}{5} + \frac{7}{5} \cdot 0 & - \frac{7}{5} + \frac{7}{5} \cdot 1 & - \frac{1}{5} + \frac{7}{5} \cdot \frac{13}{215} & 0 + \frac{7}{5} \cdot \frac{1}{215} & 0 + \frac{7}{5} \cdot \frac{29}{215} \\ 0 & 1 & 0 & \frac{4}{215} & \frac{34}{645} & - \frac{89}{645} \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 & - \frac{124}{1075} & \frac{7}{1075} & \frac{203}{1075} \\ 0 & 1 & 0 & \frac{4}{215} & \frac{34}{645} & - \frac{89}{645} \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Perform the row operation $R_{1} = R_{1} + \frac{6}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{6}{5} \cdot 0 & - \frac{6}{5} + \frac{6}{5} \cdot 1 & 0 + \frac{6}{5} \cdot 0 & - \frac{124}{1075} + \frac{6}{5} \cdot \frac{4}{215} & \frac{7}{1075} + \frac{6}{5} \cdot \frac{34}{645} & \frac{203}{1075} + \frac{6}{5} (- \frac{89}{645}) \\ 0 & 1 & 0 & \frac{4}{215} & \frac{34}{645} & - \frac{89}{645} \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{4}{43} & \frac{3}{43} & \frac{1}{43} \\ 0 & 1 & 0 & \frac{4}{215} & \frac{34}{645} & - \frac{89}{645} \\ 0 & 0 & 1 & \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Step 3

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} - \frac{4}{43} & \frac{3}{43} & \frac{1}{43} \\ \frac{4}{215} & \frac{34}{645} & - \frac{89}{645} \\ \frac{13}{215} & \frac{1}{215} & \frac{29}{215} \end{array}]$

Find the Inverse [[-5,6,7],[7,9,8],[2,-3,4]]

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