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If matrix $A=\left[a_{i j}\right]_{2 \times 2}$, where $a_{i j}=1$, if $i \neq j$ and 0 if $i=j$, then $A^2$ is equal to
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$I$
$I$
Since, $A=\left[a_{i j}\right]_{2 \times 2}$, where $a_{i j}=1$, if $i \neq j$ and 0 if $i=j$
So, $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$.
Hence $A^2=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I$
So, $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$.
Hence $A^2=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I$
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