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# 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

$\begin{array}{1 1}(A)\;\large\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\normalsize =0\\(B)\;\large\frac{1}{a}-\frac{1}{b}-\frac{1}{c}\normalsize =0\\(C)\;\large\frac{1}{b}+\frac{1}{c}+\frac{1}{a}\normalsize =0\\(D)\;\large\frac{1}{b}-\frac{1}{c}-\frac{1}{a}\normalsize =0\end{array}$

Given :
$\begin{vmatrix}bc&ca&ab\\ca&ab&bc\\ab&bc&ca\end{vmatrix}=0$
$\Rightarrow 3a^2b^2c^2-[(ab)^3+(bc)^3+(ca)^3]=0$
$\Rightarrow (ab+bc+ca)(a^2b^2+b^2c^2+c^2a^2-ab^2c-bc^2a-ca^2b)=0$
$\Rightarrow ab+bc+ca=0$
$\Rightarrow\large\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$=0$
Hence (A) is the correct answer.