Question

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

Answer 1

Given, $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$.

Now,

$A^2=$$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

or, $A^2=$$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I_2$ [ $I_2$ is the identity matrix of order $2$]

Now

$A^4=I_2\times I_2$

or, $A^4=I_2$.