Question

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

Answer 1

Answer: 6

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