Let the matrix A = [(1 0 0), (1 0 1), (0 1 0)] satisfy A^n = A^(n-2) + A^2 - I for n ≥ 3. Then the sum of all the elements of A^(50) is:-

Question

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 \(n \geq 3.\) Then the sum of all the elements of \(\mathrm{A}^{50}\) is :-

(1) 53

(2) 52

(3) 39

(4) 44

Answer 1

Correct option is: (1) 53 

\(A^{50}=A^{48}+A^{2}-I\)

\(=\mathrm{A}^{46}+2\left(\mathrm{~A}^{2}-\mathrm{I}\right)\)

\(=\mathrm{A}^{44}+3\left(\mathrm{~A}^{2}-\mathrm{I}\right)\)

\(=\mathrm{A}^{2}+24\left(\mathrm{~A}^{2}-\mathrm{I}\right)\)

\(=25 \mathrm{~A}^{2}-24 \mathrm{I}\)

\(=25\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]-24\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\)

\(=\left[\begin{array}{ccc}1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1\end{array}\right]\)

Sum = 53