Question

If $a,b,c$ are non-zero real numbers and if the equations $(a-1)x=y+z,(b-1)y=z+x,(c-1)z=x+y$ have a non-trivial solution then $ab+bc+ca$ equals

• A. $a+b+c$
• B. $abc$
• C. $1$
• D. none of these

Answer 1

The correct option is B $abc$

Given equations $(a-1)x-y-z=0,x-(b-1)y+z=0,x+y-(c-1)z=0$ can be written as $AX=O$
where$A=\left[\begin{array}{ccc}a-1& -1& -1\\ 1& -(b-1)& 1\\ 1& 1& -(c-1)\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],O=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
For non-trivial solution
$|A|=0$
$\left|\begin{array}{ccc}a-1& -1& -1\\ 1& -(b-1)& 1\\ 1& 1& -(c-1)\end{array}\right|=0$
$\Rightarrow (a-1)[bc-(b+c\left)\right]+b+c=abc-ab-ac-bc-b-c+b+c=0$
$\Rightarrow abc=ab+bc+ac$