https://file.aiquickdraw.com/mathbot/file/172586535922485cso7aj.jpg

The matrix you provided is:

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

### To find the eigenvalues and eigenvectors:
We follow the process of solving the characteristic equation:

1. **Find the characteristic equation:**

The characteristic equation is obtained by solving \( \det(A - \lambda I) = 0 \), where \( \lambda \) represents the eigenvalue and \( I \) is the identity matrix.

2. **Solve for the eigenvalues \( \lambda \):**

Once the determinant is computed, it gives a polynomial in terms of \( \lambda \). Solving this will give the eigenvalues.

3. **Find the eigenvectors:**

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

Let me compute this step-by-step.The eigenvalues of the matrix \( A \) are:

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

The corresponding eigenvectors are:

\[
\mathbf{v_1} = \begin{bmatrix} 0.3015 \\ 0.9045 \\ 0.3015 \end{bmatrix}, \quad
\mathbf{v_2} = \begin{bmatrix} -0.8018 \\ -0.5345 \\ -0.2673 \end{bmatrix}, \quad
\mathbf{v_3} = \begin{bmatrix} 0.7071 \\ 0 \\ 0.7071 \end{bmatrix}
\]

Would you like any further details or clarification?

Here are 5 related questions to deepen your understanding:

1. How is the characteristic equation derived from a matrix?
2. Why are eigenvectors important in linear algebra?
3. What are the properties of eigenvalues and eigenvectors in symmetric matrices?
4. How do eigenvalues affect the stability of systems in control theory?
5. Can eigenvectors be scaled, and what does that mean for the system?

**Tip:** Eigenvectors are unique up to a scalar multiple. So, any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.

Find eigenvalues and eigenvectors for the matrix provided.

This problem focuses on finding the eigenvalues and eigenvectors of a 3x3 matrix using the characteristic equation det(A - λI) = 0. It includes detailed step-by-step instructions for solving the eigenvalue problem and identifying the corresponding eigenvectors. Suitable for undergraduate-level linear algebra students.

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