Question

If A = $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, B = $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and C = $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, then show that A² = B² = C² = I₂.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}1+0& 0+0\\ 0+0& 0+1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]...\left(1\right) \\ {B}^2=BB \\ \Rightarrow B^2=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right] \\ \Rightarrow B^2=\left[\begin{array}{cc}1+0& 0-0\\ 0-0& 0+1\end{array}\right] \\ \Rightarrow B^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]...\left(2\right) \\ {C}^2=CC \\ \Rightarrow B^2=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right] \\ \Rightarrow B^2=\left[\begin{array}{cc}0+1& 0+0\\ 0+0& 1+0\end{array}\right] \\ \Rightarrow B^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]...\left(3\right) \\ \mathrm{We}\mathrm{know}, \\ {I}_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]...\left(4\right) \end{gathered}$$

$\Rightarrow A^2=B^2=C^2=I_2\left[\mathrm{From}\mathrm{eqs}.\left(1\right),\left(2\right),\left(3\right)\mathrm{and}\left(4\right)\right]$