Question

Suppose 𝐴 and 𝑃 are 3 × 3 matrices and 𝑃^-1 exists. If 𝑃^-1𝐴𝑃 = ( 1 2 3

0 4 5

0 0 6 ) then find all eigen values of 𝐴² .

Answer 1

Let $=A=\begin{bmatrix} 1&2& 3 \\ 0 & 4 & 5\\0 &0&6 \end{bmatrix}$

A $=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5\\0 & 0 & 6 \end{bmatrix}$ $=1\begin{vmatrix} 4 & 5 \\ 0 & 6 \end{vmatrix}-2\begin{vmatrix} 0&5\\ 0 & 6 \end{vmatrix}+3\begin{vmatrix} 0 & 4 \\ 0 & 0 \end{vmatrix}$

$=1(24-0)-2(0-0)+3(0-0)$

A$=24\mathrlap{\,/}{=}0$