Evaluate Expression Involving Elements of Matrix Difference

To find the value of $\frac{b_{31} - b_{21}}{b_{32}}$, we decompose matrix $A$ as $A = I + N$, where $N = \begin{bmatrix} 0 & 0 & 0 \\ 3 & 0 & 0 \\ 9 & 3 & 0 \end{bmatrix}$ is a nilpotent matrix satisfying $N^3 = 0$.

By binomial expansion, $A^{99} = (I + N)^{99} = I + 99N + \frac{99 \times 98}{2}N^2$, which simplifies $B = A^{99} - I$ to $B = 99N + 4851N^2$.

Evaluating the matrix elements yields: $b_{21} = 99 \times 3 = 297$ $b_{32} = 99 \times 3 = 297$ $b_{31} = 99 \times 9 + 4851 \times 9 = 44550$

Substituting these into the given expression gives: $\frac{b_{31} - b_{21}}{b_{32}} = \frac{44550 - 297}{297} = \frac{44253}{297} = 149$

Hence, the correct option is C.