Homogeneous Equations

Algebra Level 4

$$(a+\alpha )x+\alpha y+\alpha z=0\\ \\ \alpha x+(\alpha +b)y+\alpha z=0\\ \\ \alpha x+\alpha y+(\alpha +c)z=0$$

Given that a,b,c are non-zero real numbers satisfying the system of equations above.

The given system of equations have a non-trivial solution if $${ \alpha }^{ -1 }=k({ a }^{ -1 }+{ b }^{ -1 }+{ c }^{ -1 })$$ then what is the the value of 'k'?

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