প্রমাণ করো
`|[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)`

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)

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, z, y],[z, z+x, x], [y, x, x+y]|= 4 xyz

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

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

Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]| =x y z (x-y)(y-z)(z-x)(x+y+z)

Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]| =x y z (x-y)(y-z)(z-x)(x+y+z)

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

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