https://file.aiquickdraw.com/mathbot/file/17299387249405y2dzjiq.pdf

Your assignment from *Matrix Analysis* includes the following problems:

1. **Find the eigenvalues and eigenvectors** of the matrix
   \[
   A = \begin{bmatrix} 0 & 1 & -1 \\ -2 & 2 & 0 \\ -2 & 1 & 1 \end{bmatrix}
   \]

2. **Find the eigenvalues and eigenvectors** of the matrix
   \[
   A = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 3 \end{bmatrix}
   \]

3. **Diagonalize** the matrix
   \[
   A = \begin{bmatrix} 2 & -3 & 0 \\ 2 & -5 & 0 \\ 0 & 0 & 3 \end{bmatrix}
   \]

4. **Diagonalize the matrix** \( A^3 \) for
   \[
   A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end{bmatrix}
   \]

Would you like detailed solutions for each problem? 

Here are 5 questions to extend your understanding:

1. How does finding eigenvalues help in matrix diagonalization?
2. What is the significance of eigenvectors in a system of linear equations?
3. Can any matrix be diagonalized? Why or why not?
4. How does the diagonalization of \( A^3 \) differ from that of \( A \)?
5. What is the role of similarity transformations in matrix diagonalization?

**Tip:** When finding eigenvalues, always verify that the determinant calculation is correct to ensure accuracy.

Step-by-step solutions for finding eigenvalues and eigenvectors, diagonalizing matrices, and matrix exponentiation. Key topics in linear algebra and matrix theory covered, including the diagonalization theorem. Suitable for undergraduate students.

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