Let I be an identity matrix of order
Question:

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

Solution:

$P=\left[\begin{array}{ll}2 & -1 \\ 5 & -3\end{array}\right]$

$5 I-8 P=\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right]-\left[\begin{array}{cc}16 & -8 \\ 40 & -24\end{array}\right]=\left[\begin{array}{ll}-11 & 8 \\ -40 & 29\end{array}\right]$

$P^{2}=\left[\begin{array}{ll}-1 & 1 \\ -5 & 4\end{array}\right]$

$P^{3}=\left[\begin{array}{cc}3 & -2 \\ 10 & -7\end{array}\right] \Rightarrow P^{6}=\left[\begin{array}{ll}-11 & 8 \\ -40 & 29\end{array}\right]=\mathrm{P}^{n}$

$\Rightarrow \mathrm{n}=6$