Find the Inverse [[4,-3,5,2,4],[1,6,3,-5,-3],[7,3,-4,-4,-5],[2,-1,8,-6,7],[6,3,-5,5,-2]]

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

Step 1

Set up a $5 \times 10$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

Step 2

Find the reduced row echelon form.

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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

Simplify $R_{1}$ .

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

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

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

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{27}{4} & \frac{7}{4} & - \frac{11}{2} & - 4 & - \frac{1}{4} & 1 & 0 & 0 & 0 \\ 7 & 3 & - 4 & - 4 & - 5 & 0 & 0 & 1 & 0 & 0 \\ 2 & - 1 & 8 & - 6 & 7 & 0 & 0 & 0 & 1 & 0 \\ 6 & 3 & - 5 & 5 & - 2 & 0 & 0 & 0 & 0 & 1 \end{array}]$

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

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{27}{4} & \frac{7}{4} & - \frac{11}{2} & - 4 & - \frac{1}{4} & 1 & 0 & 0 & 0 \\ 7 - 7 \cdot 1 & 3 - 7 (- \frac{3}{4}) & - 4 - 7 (\frac{5}{4}) & - 4 - 7 (\frac{1}{2}) & - 5 - 7 \cdot 1 & 0 - 7 (\frac{1}{4}) & 0 - 7 \cdot 0 & 1 - 7 \cdot 0 & 0 - 7 \cdot 0 & 0 - 7 \cdot 0 \\ 2 & - 1 & 8 & - 6 & 7 & 0 & 0 & 0 & 1 & 0 \\ 6 & 3 & - 5 & 5 & - 2 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Simplify $R_{3}$ .

Perform the row operation $R_{4} = R_{4} - 2 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{27}{4} & \frac{7}{4} & - \frac{11}{2} & - 4 & - \frac{1}{4} & 1 & 0 & 0 & 0 \\ 0 & \frac{33}{4} & - \frac{51}{4} & - \frac{15}{2} & - 12 & - \frac{7}{4} & 0 & 1 & 0 & 0 \\ 2 - 2 \cdot 1 & - 1 - 2 (- \frac{3}{4}) & 8 - 2 (\frac{5}{4}) & - 6 - 2 (\frac{1}{2}) & 7 - 2 \cdot 1 & 0 - 2 (\frac{1}{4}) & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 \\ 6 & 3 & - 5 & 5 & - 2 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{27}{4} & \frac{7}{4} & - \frac{11}{2} & - 4 & - \frac{1}{4} & 1 & 0 & 0 & 0 \\ 0 & \frac{33}{4} & - \frac{51}{4} & - \frac{15}{2} & - 12 & - \frac{7}{4} & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{11}{2} & - 7 & 5 & - \frac{1}{2} & 0 & 0 & 1 & 0 \\ 6 & 3 & - 5 & 5 & - 2 & 0 & 0 & 0 & 0 & 1 \end{array}]$

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

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{27}{4} & \frac{7}{4} & - \frac{11}{2} & - 4 & - \frac{1}{4} & 1 & 0 & 0 & 0 \\ 0 & \frac{33}{4} & - \frac{51}{4} & - \frac{15}{2} & - 12 & - \frac{7}{4} & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{11}{2} & - 7 & 5 & - \frac{1}{2} & 0 & 0 & 1 & 0 \\ 6 - 6 \cdot 1 & 3 - 6 (- \frac{3}{4}) & - 5 - 6 (\frac{5}{4}) & 5 - 6 (\frac{1}{2}) & - 2 - 6 \cdot 1 & 0 - 6 (\frac{1}{4}) & 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & 1 - 6 \cdot 0 \end{array}]$

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{27}{4} & \frac{7}{4} & - \frac{11}{2} & - 4 & - \frac{1}{4} & 1 & 0 & 0 & 0 \\ 0 & \frac{33}{4} & - \frac{51}{4} & - \frac{15}{2} & - 12 & - \frac{7}{4} & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{11}{2} & - 7 & 5 & - \frac{1}{2} & 0 & 0 & 1 & 0 \\ 0 & \frac{15}{2} & - \frac{25}{2} & 2 & - 8 & - \frac{3}{2} & 0 & 0 & 0 & 1 \end{array}]$

Multiply each element of $R_{2}$ by $\frac{4}{27}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ \frac{4}{27} \cdot 0 & \frac{4}{27} \cdot \frac{27}{4} & \frac{4}{27} \cdot \frac{7}{4} & \frac{4}{27} (- \frac{11}{2}) & \frac{4}{27} \cdot - 4 & \frac{4}{27} (- \frac{1}{4}) & \frac{4}{27} \cdot 1 & \frac{4}{27} \cdot 0 & \frac{4}{27} \cdot 0 & \frac{4}{27} \cdot 0 \\ 0 & \frac{33}{4} & - \frac{51}{4} & - \frac{15}{2} & - 12 & - \frac{7}{4} & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{11}{2} & - 7 & 5 & - \frac{1}{2} & 0 & 0 & 1 & 0 \\ 0 & \frac{15}{2} & - \frac{25}{2} & 2 & - 8 & - \frac{3}{2} & 0 & 0 & 0 & 1 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & \frac{33}{4} & - \frac{51}{4} & - \frac{15}{2} & - 12 & - \frac{7}{4} & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{11}{2} & - 7 & 5 & - \frac{1}{2} & 0 & 0 & 1 & 0 \\ 0 & \frac{15}{2} & - \frac{25}{2} & 2 & - 8 & - \frac{3}{2} & 0 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{3} = R_{3} - \frac{33}{4} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 - \frac{33}{4} \cdot 0 & \frac{33}{4} - \frac{33}{4} \cdot 1 & - \frac{51}{4} - \frac{33}{4} \cdot \frac{7}{27} & - \frac{15}{2} - \frac{33}{4} (- \frac{22}{27}) & - 12 - \frac{33}{4} (- \frac{16}{27}) & - \frac{7}{4} - \frac{33}{4} (- \frac{1}{27}) & 0 - \frac{33}{4} \cdot \frac{4}{27} & 1 - \frac{33}{4} \cdot 0 & 0 - \frac{33}{4} \cdot 0 & 0 - \frac{33}{4} \cdot 0 \\ 0 & \frac{1}{2} & \frac{11}{2} & - 7 & 5 & - \frac{1}{2} & 0 & 0 & 1 & 0 \\ 0 & \frac{15}{2} & - \frac{25}{2} & 2 & - 8 & - \frac{3}{2} & 0 & 0 & 0 & 1 \end{array}]$

Simplify $R_{3}$ .

Perform the row operation $R_{4} = R_{4} - \frac{1}{2} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & - \frac{134}{9} & - \frac{7}{9} & - \frac{64}{9} & - \frac{13}{9} & - \frac{11}{9} & 1 & 0 & 0 \\ 0 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & \frac{11}{2} - \frac{1}{2} \cdot \frac{7}{27} & - 7 - \frac{1}{2} (- \frac{22}{27}) & 5 - \frac{1}{2} (- \frac{16}{27}) & - \frac{1}{2} - \frac{1}{2} (- \frac{1}{27}) & 0 - \frac{1}{2} \cdot \frac{4}{27} & 0 - \frac{1}{2} \cdot 0 & 1 - \frac{1}{2} \cdot 0 & 0 - \frac{1}{2} \cdot 0 \\ 0 & \frac{15}{2} & - \frac{25}{2} & 2 & - 8 & - \frac{3}{2} & 0 & 0 & 0 & 1 \end{array}]$

Simplify $R_{4}$ .

Perform the row operation $R_{5} = R_{5} - \frac{15}{2} R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & - \frac{134}{9} & - \frac{7}{9} & - \frac{64}{9} & - \frac{13}{9} & - \frac{11}{9} & 1 & 0 & 0 \\ 0 & 0 & \frac{145}{27} & - \frac{178}{27} & \frac{143}{27} & - \frac{13}{27} & - \frac{2}{27} & 0 & 1 & 0 \\ 0 - \frac{15}{2} \cdot 0 & \frac{15}{2} - \frac{15}{2} \cdot 1 & - \frac{25}{2} - \frac{15}{2} \cdot \frac{7}{27} & 2 - \frac{15}{2} (- \frac{22}{27}) & - 8 - \frac{15}{2} (- \frac{16}{27}) & - \frac{3}{2} - \frac{15}{2} (- \frac{1}{27}) & 0 - \frac{15}{2} \cdot \frac{4}{27} & 0 - \frac{15}{2} \cdot 0 & 0 - \frac{15}{2} \cdot 0 & 1 - \frac{15}{2} \cdot 0 \end{array}]$

Simplify $R_{5}$ .

Multiply each element of $R_{3}$ by $- \frac{9}{134}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ - \frac{9}{134} \cdot 0 & - \frac{9}{134} \cdot 0 & - \frac{9}{134} (- \frac{134}{9}) & - \frac{9}{134} (- \frac{7}{9}) & - \frac{9}{134} (- \frac{64}{9}) & - \frac{9}{134} (- \frac{13}{9}) & - \frac{9}{134} (- \frac{11}{9}) & - \frac{9}{134} \cdot 1 & - \frac{9}{134} \cdot 0 & - \frac{9}{134} \cdot 0 \\ 0 & 0 & \frac{145}{27} & - \frac{178}{27} & \frac{143}{27} & - \frac{13}{27} & - \frac{2}{27} & 0 & 1 & 0 \\ 0 & 0 & - \frac{130}{9} & \frac{73}{9} & - \frac{32}{9} & - \frac{11}{9} & - \frac{10}{9} & 0 & 0 & 1 \end{array}]$

Simplify $R_{3}$ .

Perform the row operation $R_{4} = R_{4} - \frac{145}{27} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{7}{134} & \frac{32}{67} & \frac{13}{134} & \frac{11}{134} & - \frac{9}{134} & 0 & 0 \\ 0 - \frac{145}{27} \cdot 0 & 0 - \frac{145}{27} \cdot 0 & \frac{145}{27} - \frac{145}{27} \cdot 1 & - \frac{178}{27} - \frac{145}{27} \cdot \frac{7}{134} & \frac{143}{27} - \frac{145}{27} \cdot \frac{32}{67} & - \frac{13}{27} - \frac{145}{27} \cdot \frac{13}{134} & - \frac{2}{27} - \frac{145}{27} \cdot \frac{11}{134} & 0 - \frac{145}{27} (- \frac{9}{134}) & 1 - \frac{145}{27} \cdot 0 & 0 - \frac{145}{27} \cdot 0 \\ 0 & 0 & - \frac{130}{9} & \frac{73}{9} & - \frac{32}{9} & - \frac{11}{9} & - \frac{10}{9} & 0 & 0 & 1 \end{array}]$

Simplify $R_{4}$ .

Perform the row operation $R_{5} = R_{5} + \frac{130}{9} R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{7}{134} & \frac{32}{67} & \frac{13}{134} & \frac{11}{134} & - \frac{9}{134} & 0 & 0 \\ 0 & 0 & 0 & - \frac{921}{134} & \frac{183}{67} & - \frac{403}{402} & - \frac{69}{134} & \frac{145}{402} & 1 & 0 \\ 0 + \frac{130}{9} \cdot 0 & 0 + \frac{130}{9} \cdot 0 & - \frac{130}{9} + \frac{130}{9} \cdot 1 & \frac{73}{9} + \frac{130}{9} \cdot \frac{7}{134} & - \frac{32}{9} + \frac{130}{9} \cdot \frac{32}{67} & - \frac{11}{9} + \frac{130}{9} \cdot \frac{13}{134} & - \frac{10}{9} + \frac{130}{9} \cdot \frac{11}{134} & 0 + \frac{130}{9} (- \frac{9}{134}) & 0 + \frac{130}{9} \cdot 0 & 1 + \frac{130}{9} \cdot 0 \end{array}]$

Simplify $R_{5}$ .

Multiply each element of $R_{4}$ by $- \frac{134}{921}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{7}{134} & \frac{32}{67} & \frac{13}{134} & \frac{11}{134} & - \frac{9}{134} & 0 & 0 \\ - \frac{134}{921} \cdot 0 & - \frac{134}{921} \cdot 0 & - \frac{134}{921} \cdot 0 & - \frac{134}{921} (- \frac{921}{134}) & - \frac{134}{921} \cdot \frac{183}{67} & - \frac{134}{921} (- \frac{403}{402}) & - \frac{134}{921} (- \frac{69}{134}) & - \frac{134}{921} \cdot \frac{145}{402} & - \frac{134}{921} \cdot 1 & - \frac{134}{921} \cdot 0 \\ 0 & 0 & 0 & \frac{594}{67} & \frac{224}{67} & \frac{12}{67} & \frac{5}{67} & - \frac{65}{67} & 0 & 1 \end{array}]$

Simplify $R_{4}$ .

Perform the row operation $R_{5} = R_{5} - \frac{594}{67} R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{7}{134} & \frac{32}{67} & \frac{13}{134} & \frac{11}{134} & - \frac{9}{134} & 0 & 0 \\ 0 & 0 & 0 & 1 & - \frac{122}{307} & \frac{403}{2763} & \frac{23}{307} & - \frac{145}{2763} & - \frac{134}{921} & 0 \\ 0 - \frac{594}{67} \cdot 0 & 0 - \frac{594}{67} \cdot 0 & 0 - \frac{594}{67} \cdot 0 & \frac{594}{67} - \frac{594}{67} \cdot 1 & \frac{224}{67} - \frac{594}{67} (- \frac{122}{307}) & \frac{12}{67} - \frac{594}{67} \cdot \frac{403}{2763} & \frac{5}{67} - \frac{594}{67} \cdot \frac{23}{307} & - \frac{65}{67} - \frac{594}{67} (- \frac{145}{2763}) & 0 - \frac{594}{67} (- \frac{134}{921}) & 1 - \frac{594}{67} \cdot 0 \end{array}]$

Simplify $R_{5}$ .

Multiply each element of $R_{5}$ by $\frac{307}{2108}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{7}{134} & \frac{32}{67} & \frac{13}{134} & \frac{11}{134} & - \frac{9}{134} & 0 & 0 \\ 0 & 0 & 0 & 1 & - \frac{122}{307} & \frac{403}{2763} & \frac{23}{307} & - \frac{145}{2763} & - \frac{134}{921} & 0 \\ \frac{307}{2108} \cdot 0 & \frac{307}{2108} \cdot 0 & \frac{307}{2108} \cdot 0 & \frac{307}{2108} \cdot 0 & \frac{307}{2108} \cdot \frac{2108}{307} & \frac{307}{2108} (- \frac{342}{307}) & \frac{307}{2108} (- \frac{181}{307}) & \frac{307}{2108} (- \frac{155}{307}) & \frac{307}{2108} \cdot \frac{396}{307} & \frac{307}{2108} \cdot 1 \end{array}]$

Simplify $R_{5}$ .

Perform the row operation $R_{4} = R_{4} + \frac{122}{307} R_{5}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{7}{134} & \frac{32}{67} & \frac{13}{134} & \frac{11}{134} & - \frac{9}{134} & 0 & 0 \\ 0 + \frac{122}{307} \cdot 0 & 0 + \frac{122}{307} \cdot 0 & 0 + \frac{122}{307} \cdot 0 & 1 + \frac{122}{307} \cdot 0 & - \frac{122}{307} + \frac{122}{307} \cdot 1 & \frac{403}{2763} + \frac{122}{307} (- \frac{171}{1054}) & \frac{23}{307} + \frac{122}{307} (- \frac{181}{2108}) & - \frac{145}{2763} + \frac{122}{307} (- \frac{5}{68}) & - \frac{134}{921} + \frac{122}{307} \cdot \frac{99}{527} & 0 + \frac{122}{307} \cdot \frac{307}{2108} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{4}$ .

Perform the row operation $R_{3} = R_{3} - \frac{32}{67} R_{5}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & - \frac{16}{27} & - \frac{1}{27} & \frac{4}{27} & 0 & 0 & 0 \\ 0 - \frac{32}{67} \cdot 0 & 0 - \frac{32}{67} \cdot 0 & 1 - \frac{32}{67} \cdot 0 & \frac{7}{134} - \frac{32}{67} \cdot 0 & \frac{32}{67} - \frac{32}{67} \cdot 1 & \frac{13}{134} - \frac{32}{67} (- \frac{171}{1054}) & \frac{11}{134} - \frac{32}{67} (- \frac{181}{2108}) & - \frac{9}{134} - \frac{32}{67} (- \frac{5}{68}) & 0 - \frac{32}{67} \cdot \frac{99}{527} & 0 - \frac{32}{67} \cdot \frac{307}{2108} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{3}$ .

Perform the row operation $R_{2} = R_{2} + \frac{16}{27} R_{5}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 + \frac{16}{27} \cdot 0 & 1 + \frac{16}{27} \cdot 0 & \frac{7}{27} + \frac{16}{27} \cdot 0 & - \frac{22}{27} + \frac{16}{27} \cdot 0 & - \frac{16}{27} + \frac{16}{27} \cdot 1 & - \frac{1}{27} + \frac{16}{27} (- \frac{171}{1054}) & \frac{4}{27} + \frac{16}{27} (- \frac{181}{2108}) & 0 + \frac{16}{27} (- \frac{5}{68}) & 0 + \frac{16}{27} \cdot \frac{99}{527} & 0 + \frac{16}{27} \cdot \frac{307}{2108} \\ 0 & 0 & 1 & \frac{7}{134} & 0 & \frac{12323}{70618} & \frac{8693}{70618} & - \frac{73}{2278} & - \frac{3168}{35309} & - \frac{2456}{35309} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 1 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & 0 & - \frac{1895}{14229} & \frac{1384}{14229} & - \frac{20}{459} & \frac{176}{1581} & \frac{1228}{14229} \\ 0 & 0 & 1 & \frac{7}{134} & 0 & \frac{12323}{70618} & \frac{8693}{70618} & - \frac{73}{2278} & - \frac{3168}{35309} & - \frac{2456}{35309} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Perform the row operation $R_{1} = R_{1} - R_{5}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 - 0 & - \frac{3}{4} - 0 & \frac{5}{4} - 0 & \frac{1}{2} - 0 & 1 - 1 & \frac{1}{4} + \frac{171}{1054} & 0 + \frac{181}{2108} & 0 + \frac{5}{68} & 0 - \frac{99}{527} & 0 - \frac{307}{2108} \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & 0 & - \frac{1895}{14229} & \frac{1384}{14229} & - \frac{20}{459} & \frac{176}{1581} & \frac{1228}{14229} \\ 0 & 0 & 1 & \frac{7}{134} & 0 & \frac{12323}{70618} & \frac{8693}{70618} & - \frac{73}{2278} & - \frac{3168}{35309} & - \frac{2456}{35309} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{1}$ .

Perform the row operation $R_{3} = R_{3} - \frac{7}{134} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 0 & \frac{869}{2108} & \frac{181}{2108} & \frac{5}{68} & - \frac{99}{527} & - \frac{307}{2108} \\ 0 & 1 & \frac{7}{27} & - \frac{22}{27} & 0 & - \frac{1895}{14229} & \frac{1384}{14229} & - \frac{20}{459} & \frac{176}{1581} & \frac{1228}{14229} \\ 0 - \frac{7}{134} \cdot 0 & 0 - \frac{7}{134} \cdot 0 & 1 - \frac{7}{134} \cdot 0 & \frac{7}{134} - \frac{7}{134} \cdot 1 & 0 - \frac{7}{134} \cdot 0 & \frac{12323}{70618} - \frac{7}{134} \cdot \frac{386}{4743} & \frac{8693}{70618} - \frac{7}{134} \cdot \frac{43}{1054} & - \frac{73}{2278} - \frac{7}{134} (- \frac{25}{306}) & - \frac{3168}{35309} - \frac{7}{134} (- \frac{112}{1581}) & - \frac{2456}{35309} - \frac{7}{134} \cdot \frac{61}{1054} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{3}$ .

Perform the row operation $R_{2} = R_{2} + \frac{22}{27} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 0 & \frac{869}{2108} & \frac{181}{2108} & \frac{5}{68} & - \frac{99}{527} & - \frac{307}{2108} \\ 0 + \frac{22}{27} \cdot 0 & 1 + \frac{22}{27} \cdot 0 & \frac{7}{27} + \frac{22}{27} \cdot 0 & - \frac{22}{27} + \frac{22}{27} \cdot 1 & 0 + \frac{22}{27} \cdot 0 & - \frac{1895}{14229} + \frac{22}{27} \cdot \frac{386}{4743} & \frac{1384}{14229} + \frac{22}{27} \cdot \frac{43}{1054} & - \frac{20}{459} + \frac{22}{27} (- \frac{25}{306}) & \frac{176}{1581} + \frac{22}{27} (- \frac{112}{1581}) & \frac{1228}{14229} + \frac{22}{27} \cdot \frac{61}{1054} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & \frac{1}{2} & 0 & \frac{869}{2108} & \frac{181}{2108} & \frac{5}{68} & - \frac{99}{527} & - \frac{307}{2108} \\ 0 & 1 & \frac{7}{27} & 0 & 0 & - \frac{8563}{128061} & \frac{619}{4743} & - \frac{455}{4131} & \frac{2288}{42687} & \frac{211}{1581} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & - \frac{3}{4} - \frac{1}{2} \cdot 0 & \frac{5}{4} - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & \frac{869}{2108} - \frac{1}{2} \cdot \frac{386}{4743} & \frac{181}{2108} - \frac{1}{2} \cdot \frac{43}{1054} & \frac{5}{68} - \frac{1}{2} (- \frac{25}{306}) & - \frac{99}{527} - \frac{1}{2} (- \frac{112}{1581}) & - \frac{307}{2108} - \frac{1}{2} \cdot \frac{61}{1054} \\ 0 & 1 & \frac{7}{27} & 0 & 0 & - \frac{8563}{128061} & \frac{619}{4743} & - \frac{455}{4131} & \frac{2288}{42687} & \frac{211}{1581} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & 0 & 0 & \frac{7049}{18972} & \frac{69}{1054} & \frac{35}{306} & - \frac{241}{1581} & - \frac{92}{527} \\ 0 & 1 & \frac{7}{27} & 0 & 0 & - \frac{8563}{128061} & \frac{619}{4743} & - \frac{455}{4131} & \frac{2288}{42687} & \frac{211}{1581} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Perform the row operation $R_{2} = R_{2} - \frac{7}{27} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & 0 & 0 & \frac{7049}{18972} & \frac{69}{1054} & \frac{35}{306} & - \frac{241}{1581} & - \frac{92}{527} \\ 0 - \frac{7}{27} \cdot 0 & 1 - \frac{7}{27} \cdot 0 & \frac{7}{27} - \frac{7}{27} \cdot 1 & 0 - \frac{7}{27} \cdot 0 & 0 - \frac{7}{27} \cdot 0 & - \frac{8563}{128061} - \frac{7}{27} \cdot \frac{95}{558} & \frac{619}{4743} - \frac{7}{27} \cdot \frac{15}{124} & - \frac{455}{4131} - \frac{7}{27} (- \frac{1}{36}) & \frac{2288}{42687} - \frac{7}{27} (- \frac{8}{93}) & \frac{211}{1581} - \frac{7}{27} (- \frac{9}{124}) \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{5}{4} & 0 & 0 & \frac{7049}{18972} & \frac{69}{1054} & \frac{35}{306} & - \frac{241}{1581} & - \frac{92}{527} \\ 0 & 1 & 0 & 0 & 0 & - \frac{117}{1054} & \frac{209}{2108} & - \frac{7}{68} & \frac{40}{527} & \frac{321}{2108} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Perform the row operation $R_{1} = R_{1} - \frac{5}{4} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{4} \cdot 0 & - \frac{3}{4} - \frac{5}{4} \cdot 0 & \frac{5}{4} - \frac{5}{4} \cdot 1 & 0 - \frac{5}{4} \cdot 0 & 0 - \frac{5}{4} \cdot 0 & \frac{7049}{18972} - \frac{5}{4} \cdot \frac{95}{558} & \frac{69}{1054} - \frac{5}{4} \cdot \frac{15}{124} & \frac{35}{306} - \frac{5}{4} (- \frac{1}{36}) & - \frac{241}{1581} - \frac{5}{4} (- \frac{8}{93}) & - \frac{92}{527} - \frac{5}{4} (- \frac{9}{124}) \\ 0 & 1 & 0 & 0 & 0 & - \frac{117}{1054} & \frac{209}{2108} & - \frac{7}{68} & \frac{40}{527} & \frac{321}{2108} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & 0 & 0 & 0 & \frac{6023}{37944} & - \frac{723}{8432} & \frac{365}{2448} & - \frac{71}{1581} & - \frac{707}{8432} \\ 0 & 1 & 0 & 0 & 0 & - \frac{117}{1054} & \frac{209}{2108} & - \frac{7}{68} & \frac{40}{527} & \frac{321}{2108} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Perform the row operation $R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{4} \cdot 0 & - \frac{3}{4} + \frac{3}{4} \cdot 1 & 0 + \frac{3}{4} \cdot 0 & 0 + \frac{3}{4} \cdot 0 & 0 + \frac{3}{4} \cdot 0 & \frac{6023}{37944} + \frac{3}{4} (- \frac{117}{1054}) & - \frac{723}{8432} + \frac{3}{4} \cdot \frac{209}{2108} & \frac{365}{2448} + \frac{3}{4} (- \frac{7}{68}) & - \frac{71}{1581} + \frac{3}{4} \cdot \frac{40}{527} & - \frac{707}{8432} + \frac{3}{4} \cdot \frac{321}{2108} \\ 0 & 1 & 0 & 0 & 0 & - \frac{117}{1054} & \frac{209}{2108} & - \frac{7}{68} & \frac{40}{527} & \frac{321}{2108} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & \frac{358}{4743} & - \frac{6}{527} & \frac{11}{153} & \frac{19}{1581} & \frac{16}{527} \\ 0 & 1 & 0 & 0 & 0 & - \frac{117}{1054} & \frac{209}{2108} & - \frac{7}{68} & \frac{40}{527} & \frac{321}{2108} \\ 0 & 0 & 1 & 0 & 0 & \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ 0 & 0 & 0 & 1 & 0 & \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ 0 & 0 & 0 & 0 & 1 & - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Step 3

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

$[\begin{array}{c} \frac{358}{4743} & - \frac{6}{527} & \frac{11}{153} & \frac{19}{1581} & \frac{16}{527} \\ - \frac{117}{1054} & \frac{209}{2108} & - \frac{7}{68} & \frac{40}{527} & \frac{321}{2108} \\ \frac{95}{558} & \frac{15}{124} & - \frac{1}{36} & - \frac{8}{93} & - \frac{9}{124} \\ \frac{386}{4743} & \frac{43}{1054} & - \frac{25}{306} & - \frac{112}{1581} & \frac{61}{1054} \\ - \frac{171}{1054} & - \frac{181}{2108} & - \frac{5}{68} & \frac{99}{527} & \frac{307}{2108} \end{array}]$

Find the Inverse [[4,-3,5,2,4],[1,6,3,-5,-3],[7,3,-4,-4,-5],[2,-1,8,-6,7],[6,3,-5,5,-2]]

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