Question

Let \( A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix} \). If \( B = [b_{ij}]_{3 \times 3} \) and \( B = A^{99} - I \), then find \( \frac{b_{31} - b_{21}}{b_{32}} \).

Hint:

For a matrix where sub-diagonal elements are \(k\) and the element at (3,1) is \(k^2\), the power \(A^n\) follows a predictable pattern related to arithmetic series.

Answer 1

Correct Answer: 149