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Question: Answered & Verified by Expert
If $A=\left(\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right), B=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right), C=\left(\begin{array}{ll}0 & i \\ i & 0\end{array}\right)$
where $i=\sqrt{-1}$, then which one of the following is correct?
MathematicsMatricesNDANDA 2013 (Phase 2)
Options:
  • A $A B=-C$
  • B $A B=C$
  • C $A^{2}=B^{2}=C^{2}=I$, where $I$ is the identity matrix
  • D $B A \neq C$
Solution:
1100 Upvotes Verified Answer
The correct answer is: $A B=-C$
$\mathrm{AB}=\left(\begin{array}{cc}\mathrm{i} & 0 \\ 0 & -1\end{array}\right)\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right)=\left(\begin{array}{cc}0 & -1 \\ -\mathrm{i} & 0\end{array}\right)$
$\mathrm{AB}=-\left(\begin{array}{ll}0 & \mathrm{i} \\ \mathrm{i} & 0\end{array}\right)=-\mathrm{C}$

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