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

Question

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

Aicharya · Accepted Answer

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

Let $I$ be an identity matrix of order $2 \times 2$ and $P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}$…

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