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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Matrices

NDA 2017 · Mathematics · Matrices

If A=[ array cc α & α - α & α array ] then what is AA ^T equal to...

If A=\left[cos⁡αsin⁡α−sin⁡αcos⁡α\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right] then what is{AA}^Tequalto(whereequal to (where\mathrm{A}^{\mathrm{T}} is the transpose of A)) ?

Options

  1. ANull matrix
  2. BIdentify matrix
  3. CA
  4. D-A

Correct answer

B. Identify matrix

Step-by-step solution

A=[cos⁡αsin⁡α−sin⁡αcos⁡α]AAT=[cos⁡αsin⁡α−sin⁡αcos⁡α][cos⁡α−sin⁡αsin⁡αcos⁡α]\begin{aligned} & A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right] \\ & A A^T=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]\end{aligned}=[cos⁡2α+sin⁡2α−cos⁡αsin⁡α+sin⁡αcos⁡α−sin⁡αcos⁡α+sin⁡αcos⁡αsin⁡2α+cos⁡2α]=[1001]\begin{aligned} & =\left[\begin{array}{cc}\cos ^2 \alpha+\sin ^2 \alpha & -\cos \alpha \sin \alpha+\sin \alpha \cos \alpha \\ -\sin \alpha \cos \alpha+\sin \alpha \cos \alpha & \sin ^2 \alpha+\cos ^2 \alpha\end{array}\right] \\ & \quad=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\end{aligned}
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