Step-by-Step Solution
Key Concept: Recognize that A is nilpotent (A² = 0) to simplify B = I + A and compute B^100 using the binomial expansion (I + A)^100 = I + 100A. Extract the matrix elements to find the ratio c/b = -9.
Computing successive powers of $A = \begin{pmatrix} 3 & 1 \\ -9 & -3 \end{pmatrix}$, we find that $A^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$, so $A$ is nilpotent. This gives $A^{100} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ and leads to $B^2 - 2B + I = 0$.
Correct Answer: 2