JEE Math Practice Question

If $$A = \left[ {\begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} } \right]$$ and $$I = \left[ {\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} } \right],$$ then which one of the following holds for all $$n \ge 1,$$ by the principle of mathematical induction?

1. A.$${A^n} = nA - \left( {n - 1} \right){\rm I}$$
2. B.$${A^n} = {2^{n - 1}}A - \left( {n - 1} \right){\rm I}$$
3. C.$${A^n} = nA + \left( {n - 1} \right){\rm I}$$
4. D.$${A^n} = {2^{n - 1}}A + \left( {n - 1} \right){\rm I}$$