Find the Inverse [[1,0,1],[1,1,-9],[0,1,-9]]

$[\begin{array}{c} 1 & 0 & 1 \\ 1 & 1 & - 9 \\ 0 & 1 & - 9 \end{array}]$

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.

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & - 9 & 0 & 1 & 0 \\ 0 & 1 & - 9 & 0 & 0 & 1 \end{array}]$

Step 2

Find the reduced row echelon form.

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 1 - 1 & 1 - 0 & - 9 - 1 & 0 - 1 & 1 - 0 & 0 - 0 \\ 0 & 1 & - 9 & 0 & 0 & 1 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & - 10 & - 1 & 1 & 0 \\ 0 & 1 & - 9 & 0 & 0 & 1 \end{array}]$

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & - 10 & - 1 & 1 & 0 \\ 0 - 0 & 1 - 1 & - 9 + 10 & 0 + 1 & 0 - 1 & 1 - 0 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & - 10 & - 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & - 1 & 1 \end{array}]$

Perform the row operation $R_{2} = R_{2} + 10 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 + 10 \cdot 0 & 1 + 10 \cdot 0 & - 10 + 10 \cdot 1 & - 1 + 10 \cdot 1 & 1 + 10 \cdot - 1 & 0 + 10 \cdot 1 \\ 0 & 0 & 1 & 1 & - 1 & 1 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 9 & - 9 & 10 \\ 0 & 0 & 1 & 1 & - 1 & 1 \end{array}]$

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

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & 1 - 1 & 0 + 1 & 0 - 1 \\ 0 & 1 & 0 & 9 & - 9 & 10 \\ 0 & 0 & 1 & 1 & - 1 & 1 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 1 & - 1 \\ 0 & 1 & 0 & 9 & - 9 & 10 \\ 0 & 0 & 1 & 1 & - 1 & 1 \end{array}]$

Step 3

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

$[\begin{array}{c} 0 & 1 & - 1 \\ 9 & - 9 & 10 \\ 1 & - 1 & 1 \end{array}]$

Find the Inverse [[1,0,1],[1,1,-9],[0,1,-9]]

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