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.