91Ó°ÊÓ

(A) Define the characteristic polynomial of the matrix \(\mathrm{A}\), (B) Let \(\mathrm{A}=\begin{array}{ccc}\mid 1 & 2 & -1 \mid \\ \mid 1 & 0 & 1 \mid \\ \mid 4 & -4 & 5\end{array}\) Find the characteristic polynomial of \(\mathrm{A}\).

Show that the following theorem is true: If two matrices are similar, then they have the same characteristic polynomial. Then show, by means of a counter-example, that the converse is false.