Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0\end{array} ight]$ satisfy $A^n=A^{n-2}+A^2-I$ for $\mathrm{n} \geq 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :-

Question

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0\end{array}ight]$ satisfy $A^n=A^{n-2}+A^2-I$ for $\mathrm{n} \geq 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :-

Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{A}^{50}=\mathrm{A}^{48}+\mathrm{A}^2-\mathrm{I} \\ & =\mathrm{A}^{46}+2\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =\mathrm{A}^{44}+3\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =\mathrm{A}^2+24\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =25 \mathrm{~A}^2-24 \mathrm{I}\end{aligned}$
$\begin{aligned}
& =25\left[\begin{array}{lll}
1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1
\end{array}\right]-24\left[\begin{array}{lll}
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1
\end{array}\right] \\ & =\left[\begin{array}{ccc}
1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1
\end{array}\right]
\end{aligned}$
Sum $=53$
option (1) ,

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy…

Solution