Question

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

Answer 1

Find the product

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

Now,

$A^2=A\times A$

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

Now,

$A^4=A^2\times A^2$

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

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