Q. 344.3( 9 Votes )

If x ≠ y ≠ z and

Answer :

By properties of determinants, we can split the given determinant into 2 parts



Taking x, y, z common from R1, R2, R3 respectively



Operating R1R1-R3, R2R2- R3




Taking (x-z) and (y-z) common from R1, R2



Expanding with R3


y2+yz+z2-x2-xz-z2 = xyz(xy2+xyz+xz2+zy2+yz2+z3-x2y-xyz-yz2-x2z-xz2-z3)


(y-x)(y+x) +z(y-x) =xyz(xy2+zy2 -x2y -x2z)


(y-x)(x+y+z)=xyz(xy(y-x)+z(y2-x2))


(y-x)(x+y+z)= xyz(xy(y-x)+z(x+y)(y-x))


(y-x)(x+y+z) = xyz(xy(y-x)+(xz+yz)(y-x))


(y-x)(x+y+z)= xyz(y-x)(xy+xz+yz)


x+y+z = xyz(xy+xz+yz)


Hence Proved


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