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Question
If \[\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16\] , then the value of \[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}\] is
Options
4
8
16
32
Solution
\[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix} = \begin{vmatrix}p & a & a \\ q & b & b \\ r & c & c\end{vmatrix} + \begin{vmatrix}p & a & p \\ q & b & q \\ r & c & r\end{vmatrix} + \begin{vmatrix}p & x & a \\ q & y & b \\ r & z & c\end{vmatrix} + \begin{vmatrix}p & x & p \\ q & y & q \\ r & z & r\end{vmatrix} + \begin{vmatrix}x & a & a \\ y & b & b \\ z & c & c\end{vmatrix} + \begin{vmatrix}x & a & p \\ y & b & q \\ z & c & r\end{vmatrix} + \begin{vmatrix}x & x & a \\ y & y & b \\ z & z & c\end{vmatrix} + \begin{vmatrix}x & x & p \\ y & y & q \\ z & z & r\end{vmatrix}\]
\[ = 0 + 0 + \begin{vmatrix}p & x & a \\ q & y & b \\ r & z & c\end{vmatrix} + 0 + 0 + \begin{vmatrix}x & a & p \\ y & b & q \\ z & c & r\end{vmatrix} + 0 + 0\]
\[ = \begin{vmatrix}p & x & a \\ q & y & b \\ r & z & c\end{vmatrix} + \begin{vmatrix}x & a & p \\ y & b & q \\ z & c & r\end{vmatrix}\]
\[ = 2\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix}\]
\[ = 2 \times 16 = 32\]
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