If ( x y z ) are all different and not equal to zero and ( | array cccc 1+x & 1 & 1 1 & 1+y & 1 1 & 1 & 1+z array |=0 ) then the value of ( x⁻¹+y⁻¹+z⁻¹ ) is equal to

Step-by-step solution

Given that, [ array l | array ccc 1+x & 1 & 1 1 & 1+y & 1 1 & 1 & 1+z array |=0 R₁ R₁-R₂ and R₂ R₂-R₃ | array ccc x & -y & 0 0 & y & -z 1 & 1 & 1+z array |=0 x[y(1+z)+z]+y(z)=0 x y+y z+z x+x y z=0 array ] Divide by xyz both side, we get [ array l 1 x + 1 y + 1 z +1=0 x⁻¹+y⁻¹+z⁻¹=-1 array ]