Question:

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 $A^{2}=B^{2}=C^{2}=l_{2}$.

Solution:

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)]