Step-by-Step Solution
Key Concept: Given a matrix A satisfying a characteristic equation, compute powers of A using the minimal polynomial or Cayley-Hamilton theorem. If A^100 satisfies B² - 2B + I = 0, then substituting B = A^100 yields (A^100)² - 2(A^100) + I = 0, which simplifies to A^200 - 2A^100 + I = 0.
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