Question

$\left|\begin{array}{ccc}{\cos }^{2}x& \cos x\sin x& -{\sin }^{2}x\\ \cos x& \sin x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$ then det A =

• A. 0
• B. -1
• C. 1
• D. 2

Answer 1

The correct option is C 1

$\left|\begin{array}{ccc}{\cos }^{2}x& \cos x\sin x& -{\sin }^{2}x\\ \cos x& \sin x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$
$C_1\to C_1-C_3$
$=\left|\begin{array}{ccc}1& \cos x\sin x& -{\sin }^{2}x\\ 0& \sin x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$
$=1\left({\cos }^{2}x\right)-\cos x\sin x(-\cos x\sin x)-{\sin }^{2}x(-{\sin }^{2}x)$
$={\cos }^{2}x+{\cos }^{2}x{\sin }^{2}x+{\sin }^{2}x-{\cos }^{2}x{\sin }^{2}x=1$