Question

Let $I$ be an identity matrix of order $2 \times 2$ and $P =\begin{bmatrix}2 & -1 \\ 5 & -3\end{bmatrix}$ Then the value of $n \in N$ for which $P^n =5 I -8 P$ is equal to________.

Answer 1

$P =\begin{bmatrix}2 & -1 \\ 5 & -3\end{bmatrix}$
$5 I -8 P =\begin{bmatrix}5 & 0 \\ 0 & 5\end{bmatrix}-\begin{bmatrix}16 & -8 \\ 40 & -24\end{bmatrix}=\begin{bmatrix}-11 & 8 \\ -40 & 29\end{bmatrix}$
$P ^{2}=\begin{bmatrix}-1 & 1 \\ -5 & 4\end{bmatrix}$
$P ^{3}=\begin{bmatrix}3 & -2 \\ 10 & -7\end{bmatrix} \Rightarrow P ^{6}=\begin{bmatrix}-11 & 8 \\ -40 & 29\end{bmatrix}= P ^{ n }$
$\Rightarrow n =6$