Find the inverse of the matrix \[ A=\left(\begin{array}{lll} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 2 \end{array}\right) \]

Step 1 ：Find the determinant of the matrix \(A\): \[\text{det}(A) = -1\]

Step 2 ：Since the determinant is non-zero, the inverse of \(A\) exists.

Step 3 ：Find the adjugate of the matrix \(A\): \[\text{adj}(A) = \left(\begin{array}{lll} 1 & -1 & 0 \\ -1 & -1 & 1 \\ 0 & 1 & -1 \end{array}\right)\]

Step 4 ：Calculate the inverse of \(A\) using the formula: \[A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)\]

Step 5 ：\(\boxed{A^{-1} = \left(\begin{array}{lll} -1 & 1 & 0 \\ 1 & 1 & -1 \\ 0 & -1 & 1 \end{array}\right)}\)