Let the matrix A=[[|gathered 1 0 0 1 0 1 0 1 0 gathered]]| satisfy A^n=A^n-2+A^2-I for n 3…

Question

Let the matrix $A=[[|\begin{gathered} 1 \quad  0 \quad  0 \ 1 \quad  0 \quad  1 \ 0 \quad  1 \quad  0 \end{gathered}]]|$ satisfy ${A}^{n}={A}^{n-2}+{A}^{2}-I$ for $n\geqslant 3$. Then the sum of all the elements of ${A}^{50}$ is :

Accepted Answer

$\begin{gathered} \begin{gathered} \quad A=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 1 \quad 0 \quad 1 \ 0 \quad 1 \quad 0\end{array}]]| \ {A}^{2}=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 1 \quad 0 \quad 1 \ 0 \quad 1 \quad 0\end{array}]]|[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 1 \quad 0 \quad 1 \ 0 \quad 1 \quad 0\end{array}]]|=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 1 \quad 1 \quad 0 \ 1 \quad 0 \quad 1\end{array}]]| \ {A}^{3}=A+{A}^{2}-I \ {A}^{3}=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 2 \quad 0 \quad 1 \ 1 \quad 1 \quad 0\end{array}]]| \ {A}^{4}={A}^{2}+{A}^{2}-I=2{A}^{2}-I \ {A}^{4}=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 2 \quad 1 \quad 0 \ 2 \quad 0 \quad 1\end{array}]]|{A}^{5}=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 3 \quad 0 \quad 1 \ 2 \quad 1 \quad 0\end{array}]]| \ {A}^{50}=[[|\begin{array}{ccc}1 \quad 0 \quad 0 \ 25 \quad 1 \quad 0 \ 25 \quad 0 \quad 1\end{array}]]| \end{gathered} \ =53 \end{gathered}$

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