Find the eigenvalues and eigenvectors of the matrix
A= [
1 0 −1
1 2 1
2 2 3
]

Let's solve for the eigenvalues and eigenvectors of matrix \( A \), where:

\[
A = \begin{pmatrix} 
1 & 0 & -1 \\
1 & 2 & 1 \\
2 & 2 & 3 
\end{pmatrix}
\]

### Step 1: Find the Eigenvalues

The eigenvalues \( \lambda \) of matrix \( A \) are found by solving the characteristic equation:

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

Where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue. I'll first compute this determinant.

### Step 2: Find the Eigenvectors

Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation:

\[
(A - \lambda I) \mathbf{v} = 0
\]

Let me compute these steps for you.The eigenvalues of the matrix \( A \) are:

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

The corresponding eigenvectors (normalized) are:

\[
\mathbf{v}_1 = \begin{pmatrix} -0.6667 \\ 0.3333 \\ 0.6667 \end{pmatrix}, \quad
\mathbf{v}_2 = \begin{pmatrix} -0.4082 \\ 0.4082 \\ 0.8165 \end{pmatrix}, \quad
\mathbf{v}_3 = \begin{pmatrix} 0.7071 \\ -0.7071 \\ 3.86 \times 10^{-16} \end{pmatrix}
\]

These eigenvectors correspond to the eigenvalues \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \) respectively.

Would you like to explore more details or ask questions about the process?

### Related Questions:
1. How are eigenvalues used in practical applications?
2. What are the steps to normalize an eigenvector?
3. Can a matrix have complex eigenvalues and eigenvectors?
4. How do you determine if a matrix is diagonalizable?
5. What happens if two eigenvalues are the same (degenerate eigenvalues)?

### Tip:
When working with eigenvectors, it's useful to normalize them (making their length equal to 1) to simplify further computations or interpretations.

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. This detailed step-by-step solution involves calculating the characteristic equation, determining eigenvalues, and finding the corresponding eigenvectors. Suitable for undergraduate students studying linear algebra.

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