A) \[{{A}^{2}}+{{B}^{2}}\]
B) \[{{A}^{2}}+{{B}^{2}}+2AB\]
C) \[{{A}^{2}}+{{B}^{2}}+AB-BA\]
D) None of these
Correct Answer: A
Solution :
\[AB=\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right]\,\left[ \begin{matrix} 0 & -i \\ i & 0 \\ \end{matrix} \right]=\left[ \begin{matrix} i & 0 \\ 0 & -i \\ \end{matrix} \right]\] and \[BA=\left[ \begin{matrix} 0 & -i \\ i & 0 \\ \end{matrix} \right]\,\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right]=\left[ \begin{matrix} -i & 0 \\ 0 & i \\ \end{matrix} \right]=-AB\] \[\therefore AB+BA=O\] Hence, \[{{(A+B)}^{2}}={{A}^{2}}+{{B}^{2}}\].
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