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

<p>$$\begin{aligned} &\begin{aligned} & A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right] \\ & A^2=\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right] \\ & A^3=A+A^2-I \\ & A^3=\left[\begin{array}{lll} 1 & 0 & 0 \\ 2 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right] \\ & A^4=A^2+A^2-I=2 A^2-I \\ & A^4=\left[\begin{array}{lll} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right] \text { and } A^5=\left[\begin{array}{lll} 1 & 0 & 0 \\ 3 & 0 & 1 \\ 2 & 1 & 0 \end{array}\right] \\ & A^{50}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1 \end{array}\right] \end{aligned}\\ &\text { Sum of elements }=53 \end{aligned}$$</p>