Question

If $n$ is a non-negative integer and $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ , then $A^n=$

• A. $\left[\begin{array}{cc}1& 0\\ n-1& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 0\\ n& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$

Answer 1

Answer: (C)

$A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$
$A^2=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]$
Similarly, $A^3=\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right]$
$A^4=\left[\begin{array}{cc}1& 0\\ 4& 1\end{array}\right]$ and so on
Hence , $A^n=\left[\begin{array}{cc}1& 0\\ n& 1\end{array}\right]$