Mathematical Reasoning & Induction · JEE Main previous year questions

2005 1 question

1. Q592005 · 28 Apr 2005 · Shift 1

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

   1. A.$A^n=nA-(n-1)\mathrm{I}$
   2. B.$A^n=2^{n-1}A-(n-1)\mathrm{I}$
   3. C.$A^n=nA+(n-1)\mathrm{I}$
   4. D.$A^n=2^{n-1}A+(n-1)\mathrm{I}$
   Show answer

   Answer: A