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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Matrices

NDA 2022 · Mathematics · Matrices

Paragraph: Let A= ( array ccc 0 & ^2 θ & ^2 θ ^2 θ & 0 & ^2 θ ^2 θ...

Paragraph: Let A=A=\left(0sin⁡2θcos⁡2θcos⁡2θ0sin⁡2θsin⁡2θcos⁡2θ0\begin{array}{ccc}0 & \sin ^2 \theta & \cos ^2 \theta \\ \cos ^2 \theta & 0 & \sin ^2 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 0\end{array}\right)andandA=P+QwherewherePissymmetricmatrixandis symmetric matrix andQisskew−symmetricmatrix.<br><strong>Question:</strong>Whatisis skew-symmetric matrix.<br><strong>Question:</strong> What isQ equal to ?

Options

  1. A\left(01/21/21/201/21/21/20\begin{array}{ccc}0 & 1 / 2 & 1 / 2 \\ 1 / 2 & 0 & 1 / 2 \\ 1 / 2 & 1 / 2 & 0\end{array}\right)
  2. B\left(011101110\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right)
  3. C\cos 2 \theta\left(0−1110−1−110\begin{array}{ccc}0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0\end{array}\right)
  4. D\cos 2 \theta\left(0−1/21/21/20−1/2−1/21/20\begin{array}{ccc}0 & -1 / 2 & 1 / 2 \\ 1 / 2 & 0 & -1 / 2 \\ -1 / 2 & 1 / 2 & 0\end{array}\right)

Correct answer

D. \cos 2 \theta\left(0−1/21/21/20−1/2−1/21/20\begin{array}{ccc}0 & -1 / 2 & 1 / 2 \\ 1 / 2 & 0 & -1 / 2 \\ -1 / 2 & 1 / 2 & 0\end{array}\right)

Step-by-step solution

Q=12(A−AT)=12[0−cos⁡2θcos⁡2θcos⁡2θ0−cos⁡2θ−cos⁡2θcos⁡2θ0]=cos⁡2θ[0−1/21/21/20−1/2−1/21/20]\begin{aligned} & Q=\frac{1}{2}\left(A-A^T\right)=\frac{1}{2}\left[\begin{array}{ccc}0 & -\cos 2 \theta & \cos 2 \theta \\ \cos 2 \theta & 0 & -\cos 2 \theta \\ -\cos 2 \theta & \cos 2 \theta & 0\end{array}\right] \\ & =\cos 2 \theta\left[\begin{array}{ccc}0 & -1 / 2 & 1 / 2 \\ 1 / 2 & 0 & -1 / 2 \\ -1 / 2 & 1 / 2 & 0\end{array}\right]\end{aligned}
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