Let I be an identity matrix of order $2 \times 2$ and
$\mathrm{P}=\left[\begin{array}{ll}2 & -1 \\ 5 & -3\end{array}\right] .$ Then the value of $\mathrm{n} \in \mathrm{N}$ for
which $\mathrm{P}^{\mathrm{n}}=5 \mathrm{I}-8 \mathrm{P}$ is equal to
$P=\left[\begin{array}{ll}2 & -1 \\ 5 & -3\end{array}\right]$
$5 \mathrm{I}-8 \mathrm{P}=\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right]-\left[\begin{array}{cc}16 & -8 \\ 40 & -24\end{array}\right]=\left[\begin{array}{cc}-11 & 8 \\ -40 & 29\end{array}\right]$
$\mathrm{P}^{2}=\left[\begin{array}{ll}-1 & 1 \\ -5 & 4\end{array}\right]$
$\mathrm{P}^{3}=\left[\begin{array}{cc}3 & -2 \\ 10 & -7\end{array}\right] \Rightarrow \mathrm{P}^{6}=\left[\begin{array}{cc}-11 & 8 \\ -40 & 29\end{array}\right]=\mathrm{P}^{\mathrm{n}}$
$\Rightarrow \mathrm{n}=6$