Question

Find$A^n$ if A is$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

Short answer

$A^{4n+3}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$for$n\ge 0$

Step-by-step solution

Step: 1

Given A is$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

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

Step: 2

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

Step: 3

$A^{4n+3}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$for $n\ge 0$