Question

If $\left|\begin{array}{ccc}a& p& x\\ b& q& y\\ c& r& z\end{array}\right|=16$, then the value of $\left|\begin{array}{ccc}p+x& a+x& a+p\\ q+y& b+y& b+q\\ r+z& c+z& c+r\end{array}\right|$ is

(a) 4
(b) 8
(c) 16
(d) 32

Answer 1

$$\begin{gathered} \left|\begin{array}{ccc}p+x& a+x& a+p\\ q+y& b+y& b+q\\ r+z& c+z& c+r\end{array}\right|=\left|\begin{array}{ccc}p& a& a\\ q& b& b\\ r& c& c\end{array}\right|+\left|\begin{array}{ccc}p& a& p\\ q& b& q\\ r& c& r\end{array}\right|+\left|\begin{array}{ccc}p& x& a\\ q& y& b\\ r& z& c\end{array}\right|+\left|\begin{array}{ccc}p& x& p\\ q& y& q\\ r& z& r\end{array}\right|+\left|\begin{array}{ccc}x& a& a\\ y& b& b\\ z& c& c\end{array}\right|+\left|\begin{array}{ccc}x& a& p\\ y& b& q\\ z& c& r\end{array}\right|+\left|\begin{array}{ccc}x& x& a\\ y& y& b\\ z& z& c\end{array}\right|+\left|\begin{array}{ccc}x& x& p\\ y& y& q\\ z& z& r\end{array}\right| \\ =0+0+\left|\begin{array}{ccc}p& x& a\\ q& y& b\\ r& z& c\end{array}\right|+0+0+\left|\begin{array}{ccc}x& a& p\\ y& b& q\\ z& c& r\end{array}\right|+0+0 \\ =\left|\begin{array}{ccc}p& x& a\\ q& y& b\\ r& z& c\end{array}\right|+\left|\begin{array}{ccc}x& a& p\\ y& b& q\\ z& c& r\end{array}\right| \\ =2\left|\begin{array}{ccc}a& p& x\\ b& q& y\\ c& r& z\end{array}\right| \\ =2\times 16=32 \end{gathered}$$

Hence, the correct option is (d).