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

Given : $P=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$
$P^2=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]\Rightarrow P^2=\left[\begin{array}{cc}-1& 1\\ -5& 4\end{array}\right]\Rightarrow P^2+P=I\Rightarrow P^2=I-P\Rightarrow P^3=P-P^2=2P-I\Rightarrow P^4=2P^2-P=2I-3P\Rightarrow P^5=2P-3P^2=5P-3I\Rightarrow P^6=5P^2-3P=5I-8P\therefore n=6$