prove that:`|(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=0`

Answer

N/a

Explanation

We have to prove
`|(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=0`
`LHS=|(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=1/(xyz)|(x^(2)y^(2)z^(2),xyz,xy+xz),(x^(2)yz^(2),xyz,yz+xy),(x^(2)y^(2)z,xyz,xz+yz)|`
`[ :' R_(1)toxR_(1),R_(2)toyR_(2),R_(3)tozR_(3)]`
`=1/(xyz) (xyz)^(2)|(yz, 1, xy+xz),(xz, 1, yz+xy),(xy, 1, xz+yz)|`
[taking `(xyz)` common from `C_(1)` and `C_(2)`]
`=xyz|(yz,1,xy+yz+zx),(xz,1,xy+yz+zx),(xy,1,xy+yz+zx)|[C_(3)toC_(3)+C_(1)]`
`=xyz(xy+yz+zx)|(yz,1,1),(xz,1,1),(xy,1,1)|`
[taking `(xy+yz+zx)` common from `C_(3)`]
`=0` [since `C_(2)` and `C_(3)` are identicals ]
`=RHS`