Prove `|(y+z,z+x,x+y),(z+x,x+y,y+z),(x+y,y+z,z+x)|=2|(x,y,z),(y,z,z),(z,x,y)|=-2(x^3+y^3+z^3-3xyz)`

Prove that |(x+y,y+z,z+x),(z+x,x+y,y+z),(y+z,z+x,x+y)|=2|(x,y,z),(z,x,y),(y,z,x)|

Using properties of determinants, prove that |{:(y + z ,z + x ,x + y ),(z + x ,x + y ,y + z),(x + y ,y + z,z + x ):}|=2 |{:(x, y, z),(y, z, x),(z, x, y):}|= - 2 (x^(3) + y^(3) + z^(3) - 3xyz)

Prove that |(y+z, x,y),(z+x, z, x),(x+y, y, z)| = (x+y+z)(x-z)^(2) .

det[[y+z,z+x,x+yy+z,x+y,y+zx+x,x+y,z+x]]=2det[[x,y,zy,z,zz,x,y]]=-2(x^(3)+y^(3)+z^(3)-3xyz)

The value of |(x+y,y+z,z+x),(x,y,z),(x-y,y-z,z-x)|=

Prove that : |{:(y+z,x,y),(z+x,z,x),(x+y,y,z):}|=(x+y+z)(x-z)^(2)

Prove that : |{:(y+z,x,y),(z+x,z,x),(x+y,y,z):}|=(x+y+z)(x-z)^(2)

det[[y+z,z+x,x+yy+z,x+y,y+zx+x,x+y,z+x]]=2det[[x,y,zy,z,xz,x,y]]=-2(x^(3)+y^(3)+z^(3)-3xyz)

Prove that |{:(y+z,x,x),(y,z+x,y),(z,z,x+y):}|=4xyz