Question

If $A=\left|\begin{array}{ccc}a& b& c\\ x& y& z\\ p& q& r\end{array}\right|$ and $B=\left|\begin{array}{ccc}q& -b& y\\ -p& a& -x\\ r& -c& z\end{array}\right|$, then

• A. $A=2B$
• B. $A=B$
• C. $A=-B$
• D. none of these

Answer 1

Answer: (C)

$A=-B$

$B=\left|\begin{array}{ccc}q& -b& y\\ -p& a& -x\\ r& -c& z\end{array}\right|$

$=-\left|\begin{array}{ccc}q& b& y\\ -p& -a& -x\\ r& c& z\end{array}\right|$$=\left|\begin{array}{ccc}q& b& y\\ p& a& x\\ r& c& z\end{array}\right|$

$=-\left|\begin{array}{ccc}p& a& x\\ q& b& y\\ r& c& z\end{array}\right|$$=-\left|\begin{array}{ccc}a& x& p\\ b& y& q\\ c& z& r\end{array}\right|$

$=-\left|\begin{array}{ccc}a& b& c\\ x& y& z\\ p& q& r\end{array}\right|=-A$ ($\because |A|=|A^T|$)

Hence, option C.