Question

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ then $A^{100};$

• A. $2^{100}A$
• B. $2^{99}A$
• C. $2^{101}A$
• D. None of these

Answer 1

Answer: (B)

Let $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
$A^2=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=2\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=2A$
$A^3=2^2\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],A^4=2^3\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
$A^3=2^{2}A$
$A^4=2^{3}A$
$\therefore A^n=2^{n-1}\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\Rightarrow A^{100}=2^{100-1}A$
$\therefore A^{100}=2^{99}A$