`|[y+z, x, x] , [y, z+x, y] , [z, z, x+y]|=` (i) `x^2y^2z^2` (ii) `4x^2y^2z^2` (iii) `xyz` (iv) `4xyz`

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

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

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

8x + y + 2z = 4x + 2y + z = 1x + y + z = 2

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

The determinant |[ C(x,1) ,C(x,2), C(x,3)] , [C(y,1) ,C(y,2), C(y,3)] , [C(z,1) ,C(z,2), C(z,3)]|= (i) 1/3xyz(x+y)(y+z)(z+x) (ii) 1/4xyz(x+y-z)(y+z-x) (iii) 1/12xyz(x-y)(y-z)(z-x) (iv) none

The determinant |[ C(x,1) ,C(x,2), C(x,3)] , [C(y,1) ,C(y,2), C(y,3)] , [C(z,1) ,C(z,2), C(z,3)]|= (i) 1/3xyz(x+y)(y+z)(z+x) (ii) 1/4xyz(x+y-z)(y+z-x) (iii) 1/12xyz(x-y)(y-z)(z-x) (iv) none

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

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)