Answer
$\begin{bmatrix}3&3&-4\\-2&-2&3\\-4&-7&7\end{bmatrix}$
Work Step by Step
We are given the matrix:
$B=\begin{bmatrix}1&-1&1\\2&5&-1\\2&3&0\end{bmatrix}$
Build the matrix:
$\begin{bmatrix}1&-1&1&|&1&0&0\\2&5&-1&|&0&1&0\\2&3&0&|&0&0&1\end{bmatrix}$
Multiply $R_2$ by -1 and add it to $R_3$:
$\begin{bmatrix}1&-1&1&|&1&0&0\\2&5&-1&|&0&1&0\\0&-2&1&|&0&-1&1\end{bmatrix}$
Multiply $R_1$ by -2 and add it to $R_2$:
$\begin{bmatrix}1&-1&1&|&1&0&0\\0&7&-3&|&-2&1&0\\0&-2&1&|&0&-1&1\end{bmatrix}$
Multiply $R_2$ by $\dfrac{1}{7}$:
$\begin{bmatrix}1&-1&1&|&1&0&0\\0&1&-\dfrac{3}{7}&|&-\dfrac{2}{7}&\dfrac{1}{7}&0\\0&-2&1&|&0&-1&1\end{bmatrix}$
Add $R_2$ to $R_1$:
$\begin{bmatrix}1&0&\dfrac{4}{7}&|&\dfrac{5}{7}&\dfrac{1}{7}&0\\0&1&-\dfrac{3}{7}&|&-\dfrac{2}{7}&\dfrac{1}{7}&0\\0&-2&1&|&0&-1&1\end{bmatrix}$
Multiply $R_2$ by 2 and add it to $R_3$:
$\begin{bmatrix}1&0&\dfrac{4}{7}&|&\dfrac{5}{7}&\dfrac{1}{7}&0\\0&1&-\dfrac{3}{7}&|&-\dfrac{2}{7}&\dfrac{1}{7}&0\\0&0&\dfrac{1}{7}&|&-\dfrac{4}{7}&-\dfrac{5}{7}&1\end{bmatrix}$
Multiply $R_3$ by -4 and add it to $R_1$:
$\begin{bmatrix}1&0&0&|&3&3&-4\\0&1&-\dfrac{3}{7}&|&-\dfrac{2}{7}&\dfrac{1}{7}&0\\0&0&\dfrac{1}{7}&|&-\dfrac{4}{7}&-\dfrac{5}{7}&1\end{bmatrix}$
Multiply $R_3$ by 3 and add it to $R_2$:
$\begin{bmatrix}1&0&0&|&3&3&-4\\0&1&0&|&-2&-2&3\\0&0&\dfrac{1}{7}&|&-\dfrac{4}{7}&-\dfrac{5}{7}&1\end{bmatrix}$
Multiply $R_3$ by 7:
$\begin{bmatrix}1&0&0&|&3&3&-4\\0&1&0&|&-2&-2&3\\0&0&1&|&-4&-7&7\end{bmatrix}$
The inverse of the matrix is:
$A^{-1}=\begin{bmatrix}3&3&-4\\-2&-2&3\\-4&-7&7\end{bmatrix}$
