Paragraph: Let A= ( array ccc 0 & ^2 θ & ^2 θ ^2 θ & 0 & ^2 θ ^2 θ & ^2 θ & 0 array ) and A=P+Q where P is symmetric matrix and Q is skew-symmetric matrix. Question: What is P equal to?

Correct answer: A. ( array ccc 0 & 1 / 2 & 1 / 2 1 / 2 & 0 & 1 / 2 1 / 2 & 1 / 2 & 0 array ) aligned & A=[ array ccc 0 & ^2 θ & ^2 θ ^2 θ & 0 & ^2 θ ^2 θ & ^2 θ & 0 array ] & A^T=[ array ccc 0 & ^2 θ & ^2 θ ^2 θ & 0 & ^2 θ ^2 θ & ^2 θ & 0 array ] & P=(1)/(2)(A+A^T)=[ array ccc 0 & 1 / 2 & 1 / 2 1 / 2 & 0 & 1 / 2 1 / 2 & 1 / 2 & 0 array ] aligned

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=

\left(

\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)

and

A=P+Q

where

is symmetric matrix and

is skew-symmetric matrix.<br><strong>Question:</strong> What is

P equal to?

Options

A $\left($ $\begin{array}{ccc}0 & 1 / 2 & 1 / 2 \\ 1 / 2 & 0 & 1 / 2 \\ 1 / 2 & 1 / 2 & 0\end{array}$ $\right)$
B $\left($ $\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}$ $\right)$
C $\cos 2 \theta\left($ $\begin{array}{ccc}0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0\end{array}$ $\right)$
D $\cos 2 \theta\left($ $\begin{array}{ccc}0 & -1 / 2 & 1 / 2 \\ 1 / 2 & 0 & -1 / 2 \\ -1 / 2 & 1 / 2 & 0\end{array}$ $\right)$

Correct answer

A. $\left($ $\begin{array}{ccc}0 & 1 / 2 & 1 / 2 \\ 1 / 2 & 0 & 1 / 2 \\ 1 / 2 & 1 / 2 & 0\end{array}$ $\right)$

Step-by-step solution

\begin{aligned} & \therefore A=\left[\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] \\ & A^T=\left[\begin{array}{ccc}0 & \cos ^2 \theta & \sin ^2 \theta \\ \sin ^2 \theta & 0 & \cos ^2 \theta \\ \cos ^2 \theta & \sin ^2 \theta & 0\end{array}\right] \\ & P=\frac{1}{2}\left(A+A^T\right)=\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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