Question

Find the inverse of this matrix.
1 -1 -1
-1 2 3
1 1 4

Answer 1

Let's use Gaussian elimination. Consider the augmented matrix,

`\left[\begin{array}ccc1 & -1 & -1 & 1 & 0 & 0\\-1 & 2 & 3 & 0 & 1 & 0\\1 & 1 & 4 & 0 & 0 & 1\end{array}\right]`

• Add row 1 to row 2, and add -1 (row 1) to row 3:

`\left[\begin{array}ccc1 & -1 & -1 & 1 & 0 & 0\\0 & 1 & 2 & 1 & 1 & 0\\0 & 2 & 5 & -1 & 0 & 1\end{array}\right]`

• Add -2 (row 2) to row 3:

`\left[\begin{array}ccc1 & -1 & -1 & 1 & 0 & 0\\0 & 1 & 2 & 1 & 1 & 0\\0 & 0 & 1 & -3 & -2 & 1\end{array}\right]`

• Add -2 (row 3) to row 2:

`\left[\begin{array}ccc1 & -1 & -1 & 1 & 0 & 0\\0 & 1 & 0 & 7 & 5 & -2\\0 & 0 & 1 & -3 & -2 & 1\end{array}\right]`

• Add row 2 and row 3 to row 1:

`\left[\begin{array}ccc1 & 0 & 0 & 5 & 3 & -1\\0 & 1 & 0 & 7 & 5 & -2\\0 & 0 & 1 & -3 & -2 & 1\end{array}\right]`

So the inverse is

`\begin{bmatrix}1&-1&-1\\-1&2&3\\1&1&4\end{bmatrix}^(-1) = \boxed{\begin{bmatrix}5&3&-1\\7&5&-2\\-3&-2&1\end{bmatrix}}`