show that ` [[y+x,z+x,x+y],[z+x,x+y,y+z],[x+y,y+z,z+x]] = -2(x^3+y^3+z^3-3xyz)`

show that |[y+z ,x, y],[ z+x, z, x],[x+y, y ,z]|=(x+y+z)(x-z)^2

Show that abs([y+z ,x,x],[y,z+x,y],[z,z,x+y])=4xyz

Prove that: |[y+z, z, y],[z, z+x, x], [y, x, x+y]|= 4 xyz

Prove that Det [[x + y + 2z, x, y], [z, y + z + 2x, y], [z, x, z + x + 2y]] = 2 (x + y + z) ^ 3

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)

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)

By using properties of determinants, show that : |[x+y+2z,x,y],[z,y+z+2x,y],[z,z,z+x+2y]| = 2(x+y+z)^3

Show that: |[x-y-z,2x,2x],[2y,y-z-x,2y],[2z,2z,z-x-y]|=(x+y+z)^3

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)