If $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]$ and $C=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ then show that $A^{2}=B^{2}=C^{2}=l_{2}$.
Here,
$A^{2}=A A$
$\Rightarrow A^{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{ll}1+0 & 0+0 \\ 0+0 & 0+1\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ ...(1)
$B^{2}=B B$
$\Rightarrow B^{2}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$
$\Rightarrow B^{2}=\left[\begin{array}{ll}1+0 & 0-0 \\ 0-0 & 0+1\end{array}\right]$
$\Rightarrow B^{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \quad \ldots(2)$
$C^{2}=C C$
$\Rightarrow B^{2}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
$\Rightarrow B^{2}=\left[\begin{array}{ll}0+1 & 0+0 \\ 0+0 & 1+0\end{array}\right]$
$\Rightarrow B^{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ ...(3)
We know,
$I_{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ ...(4)
$\Rightarrow A^{2}=B^{2}=C^{2}=I_{2}$ [From eqs. (1), (2), (3) and (4)]
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