Question

Find the value of the determinant $\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$, where $a,b,c$ are, respectively, the $pth,qth,rth$ terms of a harmonic progression.

Answer 1

Given $a,b$ and $c$ are respectively the $p^{th},q^{th}$ and $r^{th}$ terms of a harmonic progression.
$\Rightarrow a=\frac{1}{A+(p-1)D},b=\frac{1}{A+(q-1)D}$ and $c=\frac{1}{A+(r-1)D}$
where $A,D$ are first term and common difference of corresponding Arithmetic progression.
$\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|=abc\left|\begin{array}{ccc}1/a& 1/b& 1/c\\ p& q& r\\ 1& 1& 1\end{array}\right|$
$=abc\left|\begin{array}{ccc}A+(p-1)D& A+(q-1)D& A+(r-1)D\\ p& q& r\\ 1& 1& 1\end{array}\right|$
applying $R_1\to R_1-A{R}_3$ gives
$=abcD\left|\begin{array}{ccc}p-1& q-1& r-1\\ p& q& r\\ 1& 1& 1\end{array}\right|$
applying $R_1+R_3$ gives
$=abcD\left|\begin{array}{ccc}p& q& r\\ p& q& r\\ 1& 1& 1\end{array}\right|=0$
$\therefore \left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|=0$