Question

$\left|\begin{array}{ccc}0& p-q& p-r\\ q-p& 0& q-r\\ r-p& r-q& 0\end{array}\right|$ =

• A. $0$
• B. $(p-q)(q-r)(r-p)$
• C. $pqr$
• D. $3pqr$

Answer 1

The correct option is A $0$

$\left|\begin{array}{ccc}0& p-q& p-r\\ q-p& 0& q-r\\ r-p& r-q& 0\end{array}\right|$

$C_2\to C_2-C_3$

$\left|\begin{array}{ccc}0& r-q& p-r\\ q-p& r-q& q-r\\ r-p& r-q& 0\end{array}\right|=(r-q)\left|\begin{array}{ccc}0& 1& p-r\\ q-p& 1& q-r\\ r-p& 1& 0\end{array}\right|$

$r_2\to r_2-r_1$ and $r_3\to r_3-r_1$
$\Rightarrow \left|\begin{array}{ccc}0& 1& p-r\\ q-p& 0& q-p\\ r-p& 0& r-p\end{array}\right|$

So, $-1\left(\right(q-p\left)\right(r-p)-(r-p\left)\right)(q-p)$
$=-(r-p)\left(0\right)$
$=0$