Exercise 19

Let $f(x)$ be a polynomial. Prove that if $A$ and $B$ are similar matrices then $f(A)$ is similar to $f(B)$.

• Now show that similar matrices have the same characteristic polynomial.

• Show that similar matrices have the same minimal polynomial.

• Decide if these are similar.

  $$\begin{pmatrix}1&3\\ 2&3\end{pmatrix}\qquad\begin{pmatrix}4&-1\\ 1&1\end{pmatrix}$$