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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Matrices

NDA 2013 · Mathematics · Matrices

If A= ( array cc i & 0 0 & -i array ), B= ( array cc 0 & -1 1 & 0...

If A=\mathrm{A}=\left(i00−i\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right), \mathrm{B}=\left(0−110\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right), \mathrm{C}=\left(0ii0\begin{array}{cc}0 & i \\ i & 0\end{array}\right) where i=−1i=\sqrt{-1}, then which one of the following is correct?

Options

  1. AAB=−C\mathrm{AB}=-\mathrm{C}
  2. BAB=C\mathrm{AB}=\mathrm{C}
  3. CA2=B2=C2=I\mathrm{A}^2=\mathrm{B}^2=\mathrm{C}^2=\mathrm{I}, where I is the identity matrix
  4. DBA≠CB A \neq C

Correct answer

A. AB=−C\mathrm{AB}=-\mathrm{C}

Step-by-step solution

A⋅B=∣i00−i∣∣0−110∣=∣0−i−i0∣=−C\begin{aligned} \mathrm{A} \cdot \mathrm{B} & =\left|\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right|\left|\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right| \\ & =\left|\begin{array}{cc}0 & -i \\ -i & 0\end{array}\right|=-\mathrm{C}\end{aligned}
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