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The document you've uploaded appears to be a mathematics assignment focused on linear algebra. The assignment contains the following four questions, which cover topics such as eigenvalues, eigenvectors, matrix diagonalization, and matrix similarity:

1. **Eigenvalues and Eigenspaces**
   - Given the matrix \( A \):
     \[
     A = \begin{pmatrix} 
     1 & 3 & 5 \\
     0 & 2 & -3 \\
     0 & 1 & -2
     \end{pmatrix}
     \]
     Determine the eigenvalues of \( A \). For each eigenvalue, find a basis for the corresponding eigenspace. **(20 marks)**

2. **Matrix Powers Formula**
   - Given matrices \( A \) and \( P \):
     \[
     A = \begin{pmatrix} 
     8 & 18 \\
     -3 & -7 
     \end{pmatrix}, \quad 
     P = \begin{pmatrix} 
     3 & -2 \\
     -1 & 1 
     \end{pmatrix}
     \]
     By studying \( P^{-1}AP \), determine a formula for \( A^n \) for \( n \in \mathbb{N} \). **(10 marks)**

3. **Matrix Similarity**
   - Given matrices \( A \) and \( B \) where:
     \[
     A = \begin{pmatrix} 
     \lambda & 0 \\
     0 & \mu
     \end{pmatrix}, \quad 
     B = \begin{pmatrix} 
     \mu & 0 \\
     0 & \lambda
     \end{pmatrix}
     \]
     Determine which of the following statements is true:
     - a) \( A \) and \( B \) are similar, regardless of the values of \( \lambda \) and \( \mu \).
     - b) \( A \) and \( B \) are not similar, regardless of the values of \( \lambda \) and \( \mu \).
     - c) The question of whether \( A \) and \( B \) are similar depends on the values of \( \lambda \) and \( \mu \).
     Justify your answer. **(8 marks)**

4. **Diagonalizability of Matrices**
   - Let \( A \) be a square matrix with real entries, and suppose \( A \) is diagonalizable (i.e., similar to a diagonal matrix). Prove the following:
     - a) For \( r \in \mathbb{N} \), \( A^r \) is diagonalizable.
     - b) If \( A \) is invertible, then \( A^{-1} \) is diagonalizable.
     - c) The matrix \( A^T \) (the transpose of \( A \)) is diagonalizable. **(12 marks)**

Would you like to proceed with solutions to these questions? If so, I can start with the first one and move step by step. Let me know if you'd like further clarification on any of the problems or their sub-parts.

This linear algebra assignment covers key topics like eigenvalues, eigenvectors, matrix diagonalization, and matrix similarity. It includes detailed problems on eigenvalues and eigenspaces, matrix powers, similarity of matrices, and the diagonalizability of square matrices. Suitable for undergraduate students studying advanced mathematics.

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