Question

If $a, b, c$ are non-zero real numbers, then $\begin{vmatrix}bc&ca&ab\\ ca&ab&bc\\ ab&bc&ca\end{vmatrix}$ vanishes, when

• A. $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$
• B. $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}=0$
• C. $\frac{1}{b}+\frac{1}{c}-\frac{1}{a}=0$
• D. $\frac{1}{b}-\frac{1}{c}-\frac{1}{a}=0$

Answer 1

Answer: (A)

We have $\begin{vmatrix}bc&ca&ab\\ ca&ab&bc\\ ab&bc&ca\end{vmatrix}=0$
$\Rightarrow 3 a^{2} b^{2} c^{2}-\left[(ab)^{3}+(bc)^{3}+(ca)^{3}\right]=0$
$\Rightarrow (ab)^{3}+(bc)^{3}+(ca)^{3}-3a^{2} b^{2} c^{2}=0$
$\Rightarrow (ab+bc+ca)\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}-a b^{2} c-b c^{2} a-c a^{2} b\right)=0$
$\Rightarrow ab+bc+ca=0$
$ \Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$