Find the Inverse [[1,2,4,5],[0,1,4,-6],[0,0,1,-2],[0,0,0,-1]]

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

Step 1

Set up a $4 \times 8$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

Step 2

Find the reduced row echelon form.

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

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

Simplify $R_{4}$ .

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

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

$[\begin{array}{c} 1 & 2 & 4 & 5 & 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & - 6 & 0 & 1 & 0 & 0 \\ 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & 0 + 2 \cdot - 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 1 \end{array}]$

Simplify $R_{3}$ .

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

Perform the row operation $R_{2} = R_{2} + 6 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & 4 & 5 & 1 & 0 & 0 & 0 \\ 0 + 6 \cdot 0 & 1 + 6 \cdot 0 & 4 + 6 \cdot 0 & - 6 + 6 \cdot 1 & 0 + 6 \cdot 0 & 1 + 6 \cdot 0 & 0 + 6 \cdot 0 & 0 + 6 \cdot - 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 1 \end{array}]$

Simplify $R_{2}$ .

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

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

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

Simplify $R_{1}$ .

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

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

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

Simplify $R_{2}$ .

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

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

$[\begin{array}{c} 1 - 4 \cdot 0 & 2 - 4 \cdot 0 & 4 - 4 \cdot 1 & 0 - 4 \cdot 0 & 1 - 4 \cdot 0 & 0 - 4 \cdot 0 & 0 - 4 \cdot 1 & 5 - 4 \cdot - 2 \\ 0 & 1 & 0 & 0 & 0 & 1 & - 4 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 1 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & 0 & 1 & 0 & - 4 & 13 \\ 0 & 1 & 0 & 0 & 0 & 1 & - 4 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 1 \end{array}]$

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 0 - 2 \cdot 1 & - 4 - 2 \cdot - 4 & 13 - 2 \cdot 2 \\ 0 & 1 & 0 & 0 & 0 & 1 & - 4 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 1 \end{array}]$

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 1 & - 2 & 4 & 9 \\ 0 & 1 & 0 & 0 & 0 & 1 & - 4 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & - 1 \end{array}]$

Step 3

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

$[\begin{array}{c} 1 & - 2 & 4 & 9 \\ 0 & 1 & - 4 & 2 \\ 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & - 1 \end{array}]$

Find the Inverse [[1,2,4,5],[0,1,4,-6],[0,0,1,-2],[0,0,0,-1]]

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