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

$\begin{array}{rcl}\mathrm{A}& =& \left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right], \mathrm{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ \mathrm{Let} \mathrm{n}& =& 2, {\mathrm{A}}^2=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right] \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ & =& \left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]\\ \mathrm{and} 2\mathrm{A}-\mathrm{I}& =& \left[\begin{array}{cc}2& 0\\ 2& 2\end{array}\right]- \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ & =& \left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]\\ \therefore {\mathrm{A}}^{\mathrm{n}}& =& \mathrm{A}-(\mathrm{n}-1) \mathrm{I}\end{array}$