Question

If $p+q+r=0,$ prove that
$\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|=pqr\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|$

Answer 1

By expansion

$L.H.S.=pa(qr{a}^2-p^{2}bc)-qb(q^{2}ac-pr{b}^2)+re(pq{c}^2-r^{2}ab)$

$=pqr(a^3+b^3+c^3)-abc(p^3+q^3+r^3)$ ..$\left(1\right)$

Now $p+q=r=0$ $\therefore p^3+b^3+c^3-3abc)$

Putting in $\left(1\right)$, we get

$\Delta =pqr(a^3+b^3+c^3-3abc)$

Again $R.H.S.$

$=pqr\left[a\right(a^2-bc)-b(ca-b^2)+c(c^2-ab\left)\right]$

$=pqr[a^3+b^3+c^3-3abc]=L.H.S.$