To diagonalize the matrix $A$, we need to follow these steps:

1. Find the eigenvalues of $A$.
2. Find the eigenvectors corresponding to each eigenvalue.
3. Form the matrix $P$ using the eigenvectors as columns.
4. Form the diagonal matrix $D$ using the eigenvalues on the diagonal.
5. Verify that $A$ can be diagonalized by checking if $A = PDP^{-1}$.

Let's proceed step by step.

Step 1: Find the eigenvalues of $A$

Given matrix $A$:

[ A = \begin{pmatrix} 2 & 1 & -1 \ 1 & 1 & -2 \ -2