Question

If $bc+qr=ca+rp=ab+pq=-1$, show that
$\left|\begin{array}{ccc}ap& a& p\\ bq& b& q\\ cr& c& r\end{array}\right|=0$

Answer 1

Rewriting the given equations, we have
$1+bc+qr=0$
$1+ca+rp=0$
$1+ab+pq=0$
Multiply $1st$ by ap, $2nd$ by bq and $3rd$ by cr, we get
$ap+\left(abc\right)p+\left(pqr\right)a=0$
$bq+\left(abc\right)q+\left(pqr\right)b=0$
$cr+\left(abc\right)r+\left(pqr\right)c=0$
Put abc = x and pqr = y
$\therefore$ the above equations are
$px+ay+ap=0$
$qx+by+bq=0$
$rx+cy+cr=0$
These equations in x and y will be consistent if $D=0$
or $\left|\begin{array}{ccc}p& a& ap\\ q& b& bq\\ r& c& cr\end{array}\right|=0$
Interchange $C_1$ and $C_3$
$\therefore \left|\begin{array}{ccc}ap& a& p\\ bq& b& q\\ cr& c& r\end{array}\right|=0$