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Question:

If $\left|\begin{array}{lll}a & p & x \\ b & q & y \\ c & r & z\end{array}\right|=16$, 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

Solution:

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

$=0+0+\left|\begin{array}{lll}p & x & a \\ q & y & b \\ r & z & c\end{array}\right|+0+0+\left|\begin{array}{lll}x & a & p \\ y & b & q \\ z & c & r\end{array}\right|+0+0$

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

$=2\left|\begin{array}{lll}a & p & x \\ b & q & y \\ c & r & z\end{array}\right|$

$=2 \times 16=32$

Hence, the correct option is (d).