The problem asks to find the inverse of matrix $A = \begin{pmatrix} 1 & 1 & -1 \\ 1 & 3 & 1 \\ 2 & 2 & -1 \end{pmatrix}$.

To find the inverse of a 3x3 matrix, follow these steps:

1. Calculate the determinant of matrix $A$: $\text{det}(A) = 1 \cdot \begin{vmatrix} 3 & 1 \\ 2 & -1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 1 & 3 \\ 2 & 2 \end{vmatrix}$ We will evaluate each of these 2x2 determinants.

2. Cofactor expansion and adjugate matrix: After finding the determinant, we compute the cofactors for each element of matrix $A$ to construct the adjugate matrix.

3. Divide adjugate matrix by the determinant: If the determinant is non-zero, the inverse is given by: $A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A)$

Let me compute the detailed solution for you now.The determinant of matrix $A$ is $\text{det}(A) = 2.0$, which is non-zero, so the inverse exists.

The inverse of matrix $A$ is: