Question

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

Answer 1

Given equations can be rewritten as
$bc+qr+1=0$ ......... (i)
$ca+rp+1=0$ .......... (ii)
$ab+pq+1=0$ .......... (iii)
Multiplying (i), (ii) and (iii) by ap, bq, cr respectively, we get
$\left(abc\right)p+\left(pqr\right)a+ap=0$
$\left(abc\right)q+\left(pqr\right)b+bq=0$
$\left(abc\right)r+\left(pqr\right)c+cr=0$
These equations are consistent .( Equations are three but variables abc & pqr two).
Hence $\left|\begin{array}{ccc}p& a& ap\\ q& b& bq\\ r& c& cr\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}p& q& r\\ a& b& c\\ ap& bq& cr\end{array}\right|=0$ {Interchanging Rows into columns}
$\Rightarrow (-1)\left|\begin{array}{ccc}ap& bq& cr\\ a& b& c\\ p& q& r\end{array}\right|=0$ $(R_1\leftrightarrow R_3)$
$\Rightarrow$ Hence $\left|\begin{array}{ccc}ap& bq& cr\\ a& b& c\\ p& q& r\end{array}\right|=0$