Consider the determinant = | array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array | If a₁₃=y z, a₂₃=z x, a₃₃=x y and the minors of a₁₃, a₂₃ , a₃₃ are respectively (z-y),(z-x),(y-x) then what is the value of ?

Step-by-step solution

= | array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array |a₁₃=y z, a₂₃=z x, a₃₃=x y and M₁₃=(z-y), M₂₃=(z-x), M₃₃=(y-x) Expanding the determinant along Column 3 , aligned & =a₁₃ (a₂₁ a₃₂-a₂₂ a₃₁ )-a₂₃ (a₁₁ a₃₂-a₁₂ a₃₁ )+a₃₃ (a₁₁ a₂-a₁₂ a₁ ) & =a₁₃ M₁₃-a₂₃ M₂₃+a₃₃ M₃₃ aligned where M_ i j denotes the minor of a_ i j element. aligned & =y z(z-y)-z x(z-x)+x y(y-x) & =y z(z-y)-z^2 x+z x^2+x y^2-x^2 y & =y z(z-y)+ (x y^2-z^2 x )+ (z x^2-x^2 y ) & =y z(z-y)+x (y^2-z^2 )+x^2(z-y) & =(z-y) [y z-x(y+z)+x^2 ] & =(