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 \(\frac{b_{31} - b_{21}}{b_{32}}\).
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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.