JEE Main 2014 (Online) 9th AprilQuestion 1
If $a,b,c$ are non-zero real numbers and if the system of equations $(a-1)x=y+z$, $(b-1)y=x+z$, $(c-1)z=x+y$ has a non-trivial solution, then $ab+bc+ca$ equals :
Determinant
Solution
Non-trivial solution needs $\begin{vmatrix}a-1&-1&-1\\-1&b-1&-1\\-1&-1&c-1\end{vmatrix}=0$, which expands to $abc-ab-bc-ca=0$. So $ab+bc+ca=abc$.