Question

If $X=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$, then the value $X^n$ is

• A. $\left[\begin{array}{cc}3n& -4n\\ n& -n\end{array}\right]$
• B. $\left[\begin{array}{cc}2+n& 5-n\\ n& -n\end{array}\right]$
• C. $\left[\begin{array}{cc}{3}^n& {(-4)}^n\\ 1^n& {(-1)}^n\end{array}\right]$
• D. None of these

Answer 1

The correct option is D None of these

For $n=2$

$X^2=X.X=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]=\left[\begin{array}{cc}5& -8\\ 2& -3\end{array}\right]=\left[\begin{array}{cc}2+(2\times 2)& {(-2)}^{2+1}\\ 2& 1-(2\times 2)\end{array}\right]$

For $n=3$

$X^3=X^2.X=\left[\begin{array}{cc}5& -8\\ 2& -3\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]=\left[\begin{array}{cc}7& -16\\ 3& -5\end{array}\right]=\left[\begin{array}{cc}1+(2\times 3)& {(-2)}^{3+1}\\ 3& 1-(2\times 3)\end{array}\right]$

$\Rightarrow X^n=\left[\begin{array}{cc}1+2n& {(-2)}^{n+1}\\ n& 1-2n\end{array}\right]$

So, option (D) is correct.