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To find the eigenvalues and eigenvectors of the matrix \( A \), we need to follow these steps:

### 1. **Find the Eigenvalues:**
   - The eigenvalues \(\lambda\) are found by solving the characteristic equation, which is determined from the determinant of the matrix \( A - \lambda I = 0 \), where \( I \) is the identity matrix of the same size as \( A \).

Given matrix \( A = \begin{pmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{pmatrix} \), let's compute the characteristic equation.

\[
\text{det}(A - \lambda I) = \text{det} \left( \begin{pmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \right)
\]

\[
= \text{det} \left( \begin{pmatrix} 1-\lambda & 0 & -1 \\ 1 & 2-\lambda & 1 \\ 2 & 2 & 3-\lambda \end{pmatrix} \right)
\]

Now, let's compute this determinant.

### 2. **Find the Eigenvectors:**
   - For each eigenvalue \(\lambda_i\), solve the system \((A - \lambda_i I) \mathbf{v} = 0\) to find the corresponding eigenvector \(\mathbf{v}\).

I'll perform these calculations now.### Results:

The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 1, \quad \lambda_2 = 2, \quad \lambda_3 = 3
\]

The corresponding eigenvectors are:

1. For \(\lambda_1 = 1\):
   \[
   \mathbf{v}_1 = \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}
   \]

2. For \(\lambda_2 = 2\):
   \[
   \mathbf{v}_2 = \begin{pmatrix} -1 \\ \frac{1}{2} \\ 1 \end{pmatrix}
   \]

3. For \(\lambda_3 = 3\):
   \[
   \mathbf{v}_3 = \begin{pmatrix} -\frac{1}{2} \\ \frac{1}{2} \\ 1 \end{pmatrix}
   \]

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions:
1. How do you verify that a given vector is indeed an eigenvector of a matrix?
2. Can a matrix have complex eigenvalues and eigenvectors?
3. What is the geometric significance of eigenvalues and eigenvectors?
4. How does the determinant of a matrix relate to its eigenvalues?
5. What are the conditions for a matrix to be diagonalizable?

**Tip:** Eigenvalues provide insight into the behavior of linear transformations, such as scaling along certain directions (eigenvectors). Understanding their calculation can be crucial in various fields, including physics, engineering, and machine learning.

Find the eigenvalues and eigenvectors of the matrix A = [[1, 0, -1], [1, 2, 1], [2, 2, 3]].

Learn how to find the eigenvalues and eigenvectors of a 3x3 matrix A = [[1, 0, -1], [1, 2, 1], [2, 2, 3]] using the characteristic equation and solving for eigenvectors. This step-by-step guide covers essential linear algebra concepts, suitable for undergraduate students.

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