Let I be an identity matrix of order 22 and P=pmatrix2&-15&-3pmatrix.

Question

Let $I$ be an identity matrix of order $2	imes2$ and $P=\begin{pmatrix}2&-1\5&-3\end{pmatrix}$. Then the value of $n\in\mathbb{N}$ for which $P^n=5I-8P$ is equal to

Answer: 6

Accepted Answer

The characteristic equation of $P$: $\lambda^2-(2-3)\lambda+(2(-3)-5(-1))=0\Rightarrow\lambda^2+\lambda-1=0$, so $P^2=I-P$.

Building up: $P^3=P-P^2=P-(I-P)=2P-I$; $P^4=2P^2-P=2(I-P)-P=2I-3P$; $P^5=2P-3P^2=2P-3(I-P)=5P-3I$; $P^6=5P^2-3P=5(I-P)-3P=5I-8P$.

So $n=6$.

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