Question

If $A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$ then for every integer n $A^n$ is equal to

• A. $\left[\begin{array}{cc}1+2n& 4n\\ n& 1+2n\end{array}\right]$
• B. $\left[\begin{array}{cc}1+2n& -4n\\ n& 1-2n\end{array}\right]$
• C. $\left[\begin{array}{cc}1-2n& 4n\\ n& 1+2n\end{array}\right]$
• D. none of these

Answer 1

The correct option is B $\left[\begin{array}{cc}1+2n& -4n\\ n& 1-2n\end{array}\right]$

$\begin{array}{c}A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]\\ A^2=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]\\ =\left[\begin{array}{cc}5& -8\\ 2& -3\end{array}\right]\\ A^n=\left[\begin{array}{cc}1+2n& -4n\\ n& 1-2n\end{array}\right]\\ Hence,optionBiscorrectanswer.\end{array}$