For a given matrix A = [ , * 20 c & - , & , ] which of the follow… - Vidyadip

MCQ

For a given matrix $A =$ $\left[ {\,\begin{array}{*{20}{c}}{\cos \theta }&{ - \,\sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}\,} \right]$ which of the following statement holds good?

A
$A = A^{-1}\, \forall \,\theta \, \in \,R\,$
B
$A$ is symmetric, for $\theta = (2n + 1) ,\frac{\pi }{2}\,$ $n\, \in \,I$
✓
$A$ is an orthogonal matrix for $\theta \, \in \, R$
D
$A$ is a skew symmetric, for $\theta = n\pi$ ;$ n \, \in \, I$

✓

Answer

Correct option: C.

$A$ is an orthogonal matrix for $\theta \, \in \, R$

c
Obv. $A$ is orthogonal as $a_{11}^2\,+a_{12}^2\, = 1$ $=a_{21}^2\, +a_{22}^2\, =a_{11}^2\, +a_{22}^2\,$
for skew symmetric matrix $aii$ $= 0 ==> \theta = (2n + 1)\,\frac{\pi }{2}\,$
for symmetric matrix , $A = A^T$ ==> $sin\theta = 0 ==>\theta = n\pi$
Also $adjA =$ $\left( {\,\begin{array}{*{20}{c}}{\cos \theta }&{\sin \theta }\\ { - \sin \theta }&{\cos \theta }\end{array}\,} \right)$ and $|A| = 1$ hence $A = A^{-1}$ is possible if $sin\theta$ $= 0$

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