Mathematics - Determinant Question with Solution | TestHub

Given,$\left(a-1\right)x=y+z\Rightarrow \left(a-1\right)x-y-z=0$

$\left(b-1\right)y=z+x\Rightarrow -x+\left(b-1\right)y-z=0$

$\left(c-1\right)z=y+x=-x-y+\left(c-1\right)z=0$

For a non-trivial solution

$\left|\begin{array}{ccc}a-1& -1& -1\\ -1& b-1& -1\\ -1& -1& c-1\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}1-a& 1& 1\\ 1& 1-b& 1\\ 1& 1& 1-c\end{array}\right|=0$

$\Rightarrow \left(1-a\right)\left[\left(1-b\right)\left(1-c\right)-1\right]-1\left[\left(1-c\right)-1\right]+\left[1-\left(1-b\right)\right]=0$

$\Rightarrow 1-\left(a+b+c\right)+\left(ab+bc+ca\right)-abc-1+\left(a+b+c\right)=0$

$\Rightarrow ab+bc+ca=abc$