Diagonalize the matrix
2 1 -1
1 1 -2
1-2 1
Find A?

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

### 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.

Learn how to diagonalize the matrix 2 1 -1; 1 1 -2; 1 -2 1 with a detailed step-by-step solution. This problem covers matrix diagonalization, eigenvalues, and eigenvectors. Suitable for advanced undergraduate students studying linear algebra.

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