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

Let's test for $n=2$:
$$X^2=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]=\left[\begin{array}{cc}9-4& -12+4\\ 3-1& -4+1\end{array}\right]=\left[\begin{array}{cc}5& -8\\ 2& -3\end{array}\right]$$
Test options for $n=2$:Option A: $\left[\begin{array}{cc}6& -8\\ 2& -2\end{array}\right]\ne X^2$ Option B: $\left[\begin{array}{cc}4& 3\\ 2& -2\end{array}\right]\ne X^2$ Option C: $\left[\begin{array}{cc}9& 16\\ 1& 1\end{array}\right]\ne X^2$ Since none of the options match, the answer is "None of these".(Correct general form is $X^n=\left[\begin{array}{cc}1+2n& -4n\\ n& 1-2n\end{array}\right]$).