Question

If $\left[\begin{array}{c}\cos \theta \\ -\sin \theta \\ 0\end{array}\begin{array}{c}\sin \theta \\ \cos \theta \\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ then the value of $a_{11}{A}_{11}+a_{12}{A}_{12}+a_{13}{A}_{13}=$ where $A_{11},A_{12}{A}_{13}$ are cofactors of $a_{11},a_{12},a_{13}$ respectively

• A. $-1$
• B. $1$
• C. $0$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $1$

$A_{11}=\left|\begin{array}{cc}\cos \theta & 0\\ 0& 1\end{array}\right|=\cos \theta$

$A_{12}=-\left|\begin{array}{cc}-\sin \theta & 0\\ 0& 1\end{array}\right|=\sin \theta$

$A_{13}=\left|\begin{array}{cc}-\sin \theta & \cos \theta \\ 0& 0\end{array}\right|=0$

Now,

$a_{11}{A}_{11}+a_{12}{A}_{12}+a_{13}{A}_{13}={\cos }^2\theta +{\sin }^2\theta =1$