Question

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then $A^{100}$ is equal to.

• A. $2^{100}A$
• B. $2^{99}A$
• C. $100A$
• D. $299A$

Answer 1

The correct option is B $2^{99}A$

Given, $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
$\therefore A^2=A\cdot A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
$=2\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=2A$
Now, $A^4=A^2\cdot A^2=2A\cdot 2A$
$=4A^2=4\times 2A$
$=8A=2^{3}A$
Similarly, $A^8=2^{7}A$
$\therefore A^{100}=2^{99}A$