If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then which one of the following holds for all $n \geq 1$, by the principle of mathematical induction ?

Question

( A ) $A^n = nA + (n-1)I$
( B ) $A^n = 2^{n-1} A - (n-1)I$
( C ) $A^n = 2^{n-1} A + (n-1)I$
( D ) $A^n = nA - (n-1)I$