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To find the **eigenvalues** and **eigenvectors** of the given matrix \( A \), follow these steps:

The matrix \( A \) is:

\[
A = \begin{pmatrix}
2 & 2 & 1 \\
1 & 3 & 1 \\
1 & 2 & 2
\end{pmatrix}
\]

### Step 1: Eigenvalues
To find the eigenvalues, solve the characteristic equation:

\[
\text{det}(A - \lambda I) = 0
\]

where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix. The matrix \( A - \lambda I \) is:

\[
A - \lambda I = \begin{pmatrix}
2 - \lambda & 2 & 1 \\
1 & 3 - \lambda & 1 \\
1 & 2 & 2 - \lambda
\end{pmatrix}
\]

Now, compute the determinant of \( A - \lambda I \):

\[
\text{det}(A - \lambda I) = \begin{vmatrix}
2 - \lambda & 2 & 1 \\
1 & 3 - \lambda & 1 \\
1 & 2 & 2 - \lambda
\end{vmatrix}
\]

Let me calculate this determinant.The determinant of \( A - \lambda I \) is:

\[
\text{det}(A - \lambda I) = -\lambda^3 + 7\lambda^2 - 11\lambda + 5
\]

Now, solve the characteristic equation:

\[
-\lambda^3 + 7\lambda^2 - 11\lambda + 5 = 0
\]

This will give the eigenvalues of the matrix \( A \). Let me find the solutions for \( \lambda \).The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 1, \quad \lambda_2 = 5
\]

### Step 2: Eigenvectors
To find the eigenvectors, substitute each eigenvalue back into the equation \( (A - \lambda I) \mathbf{x} = 0 \) and solve for the vector \( \mathbf{x} \).

Let me calculate the eigenvectors for each eigenvalue.The eigenvectors corresponding to the eigenvalues are:

- For \( \lambda_1 = 1 \), the eigenvectors are:
  \[
  \mathbf{v}_1 = \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}
  \]
  
- For \( \lambda_2 = 5 \), the eigenvector is:
  \[
  \mathbf{v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}
  \]

These are the eigenvalues and their corresponding eigenvectors for the matrix \( A \).

Would you like further details on any part of this process?

### Follow-up questions:
1. How is the characteristic polynomial derived from the matrix?
2. What is the significance of the eigenvalues in a matrix?
3. How do we interpret multiple eigenvectors for the same eigenvalue?
4. How do you check if a matrix is diagonalizable?
5. What are some practical applications of eigenvalues and eigenvectors?

**Tip**: Eigenvalues tell you how much a matrix stretches or shrinks vectors in certain directions. Eigenvectors indicate those special directions.

Find the characteristic roots (eigenvalues) of the matrix A.

Learn how to find the eigenvalues and eigenvectors of the matrix A. This problem involves solving the characteristic equation det(A - λI) = 0 and finding the corresponding eigenvectors. Suitable for undergraduate students studying linear algebra.

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