Question

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

• A. $\left[\begin{array}{cc}1& 0\\ 0& 50\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 0\\ 50& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 25\\ 0& 1\end{array}\right]$
• D. None of these

Answer 1

Answer: (D)

None of these

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

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

$A^4=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]A^8=\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 4& 1\end{array}\right]$

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

$\Rightarrow A^{50}=\left[\begin{array}{cc}1& 0\\ 25& 1\end{array}\right]$