Question

If $A=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]$, then $A^{100}$ is equal to

• A. $\left[\begin{array}{cc}1& 0\\ \left(\frac{1}{2}\right)\times 100& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 0\\ 25& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 0\\ 50& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& 0\\ 100& 1\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{cc}1& 0\\ \left(\frac{1}{2}\right)\times 100& 1\end{array}\right]$

Given $A=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]$

$A^2=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \frac{2}{2}& 1\end{array}\right]$

$A^3=\left[\begin{array}{cc}1& 0\\ \frac{2}{2}& 1\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \frac{3}{2}& 1\end{array}\right]$

$A^4=\left[\begin{array}{cc}1& 0\\ \frac{3}{2}& 1\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \frac{4}{2}& 1\end{array}\right]$

$A^n=\left[\begin{array}{cc}1& 0\\ \frac{n}{2}& 1\end{array}\right]$

So

$A^{100}=\left[\begin{array}{cc}1& 0\\ \frac{100}{2}& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 50& 1\end{array}\right]$