Question

If $A=\left[\begin{array}{ccc}0& i-\sin x& i-\cos x\\ sinx-i& 0& \sin x-i\\ cosx-i& -\sin x+i& 0\end{array}\right]$ then $\left|A\right|$ equals

• A. $0$
• B. $\sin x$
• C. $\cos x$
• D. $1$

Answer 1

Answer: (A)

$0$

$A=\left[\begin{array}{ccc}0& i-\sin x& i-\cos x\\ sinx-i& 0& \sin x-i\\ cosx-i& -\sin x+i& 0\end{array}\right]$
$\therefore \left|A\right|=\left|\begin{array}{ccc}0& i-\sin x& i-\cos x\\ \sin x-i& 0& \sin x-i\\ \cos x-i& i-\sin x& 0\end{array}\right|$

$=(i-\sin x)(\cos x-i)(\sin x-i)-(i-\cos x)(i-\sin x{)}^2$

$=(i-\sin x{)}^2(i-\cos x)-(i-\sin x{)}^2(i-\cos x)=0$