question_answer

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?

A)  \[{{A}^{n}}={{2}^{n-1}}A+(n-1)|\]

B)  \[{{A}^{n}}=nA+(n-1)|\]

C)  \[{{A}^{n}}={{2}^{n-1}}A-(n-1)|\]

D)  \[{{A}^{n}}=nA-(n-1)|\]

E)  None of these