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 $A$: 1 & 3 & 5 \\ 0 & 2 & -3 \\ 0 & 1 & -2 \end{pmatrix}$$ Determine the eigenvalues of $$A$$. For each eigenvalue, find a basis for the corresponding eigenspace. **(20 marks)**
2. Matrix Powers Formula
   • Given matrices $A$ and $P$: 8 & 18 \\ -3 & -7 \end{pmatrix}, \quad P = \begin{pmatrix} 3 & -2 \\ -1 & 1 \end{pmatrix}$$ 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: \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.