Question

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

• A. 53
• B. 52
• C. 39
• D. 44

Hint:

Use matrix recurrence relations and properties to compute higher powers of matrices efficiently.

Answer 1

The Correct Option is A

Using the recurrence relation \( A^n = A^{n-2} + A^2 - I \) for \( n \geq 3 \), we can compute higher powers of the matrix \( A \). By using matrix algebra, we find that the sum of the elements of \( A^{50} \) is 53.
Thus, the correct answer is \( 53 \).