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 $P^n =5 I -8 P$ is equal to ….. .

If $A = \left[ {\begin{array}{*{20}{c}}2&5\\3&7\end{array}} \right]$and $B = \left[ {\begin{array}{*{20}{c}}0&3\\4&1\end{array}} \right],$then

If $\mathrm{A}=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right],$ prove that $\mathrm{A}^{n}=\left[\begin{array}{lll}3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1}\end{array}\right], n \in \mathrm{N}$.

If $A = \left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{ – \sin \alpha }\\{\sin \alpha }&{\cos \alpha }\end{array}} \right]$and $B = \left[ {\begin{array}{*{20}{c}}{\cos \beta }&{ – \sin \beta }\\{\sin \beta }&{\cos \beta }\end{array}} \right]$, then the correct relation is

Let $A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$. Let $\alpha, \beta \in R$ be such that $\alpha A^{2}+\beta A=2 I$. Then $\alpha+\beta$ is equal to –