To diagonalize the matrix $A$, we need to find a diagonal matrix $D$ and an invertible matrix $P$ such that $A = PDP^{-1}$. The steps to diagonalize a matrix are as follows:

1. Find the eigenvalues of $A$:

   • Compute the characteristic polynomial of $A$, which is given by $\text{det}(A - \lambda I) = 0$.
   • Solve the characteristic equation to find the eigenvalues $\lambda$.

2. Find the eigenvectors of $A$:

   • For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{v} = 0$ to find the corresponding eigenvector $\mathbf{v}$.

3. Form the matrix $P$ and the diagonal matrix $D$:

   • The matrix $P$ is formed by placing the eigenvectors as columns.
   • The matrix $D$ is a diagonal matrix with the eigenvalues on the diagonal.

Let's begin by computing the eigenvalues of $A$.

The matrix $A$ is:

2 & 1 & -1 \\ 1 & 1 & -2 \\ 1 & -2 & 1 \end{pmatrix}$$ ### Step 1: Find the Eigenvalues To find the eigenvalues, we compute the determinant of $$A - \lambda I$$: $$\text{det}(A - \lambda I) = \text{det}\left(\begin{pmatrix} 2-\lambda & 1 & -1 \\ 1 & 1-\lambda & -2 \\ 1 & -2 & 1-\lambda \end{pmatrix}\right)$$ We need to compute the determinant of this matrix and solve for $$\lambda$$. Let's calculate that determinant.