Step-by-Step Solution
Key Concept: Recursive matrix relations can be reduced to characteristic equation form to determine matrix entries.
From the relation $A^{100} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + 100 \begin{pmatrix} 3 & 1 \\ -9 & -3 \end{pmatrix}$, we deduce that $B^2 - 2B + I = 0$, which is equivalent to $2^{200} - 2 \cdot 2^{100} + I = 0$, giving the constraint for matrix $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
Correct Answer: 3