Worksheet 27

1. Determine the minimal polynomial of \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) (Hint: compute $\chi_A(x)$ and use Cayley-Hamilton to focus the question.)

2. Let \(A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix}\) What are $\chi_A(x)$ and $m_A(x)$?

3. Suppose we have a block diagonal matrix \(A = \begin{pmatrix} A_1 & 0 \\ 0 & A_2 \end{pmatrix}\) What can you say about $\chi_A(x)$ vs $\chi_{A_1}(x)$ and $\chi_{A_2}(x)$? What can you say about $m_A(x)$ versus $m_{A_1}(x)$ and $m_{A_2}(x)$? (Hint: think about the case of diagonal matrices first.)