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\[n\ge 1,\]the principle of mathematical induction?     AIEEE  Solved  Paper-2005

A) \[{{A}^{n}}={{2}^{n-1}}A+(n-1)l\]

B) \[{{A}^{n}}=nA+(n-1)l\]

C) \[{{A}^{n}}={{2}^{n-1}}A+(n-1)l\]

D) \[{{A}^{n}}=nA+(n-1)l\]

Correct Answer: D

Solution :

Given, Now,        Similarly,can be verified by induction. Now, go option by option Hence, option (d) is correct.