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

y+z,x,yz+y,z,xx+y,y,z]|=(x+y+z)(x-z)^(2)

If |[y+z,x,y],[z+x,z,x],[x+y,y,z]|=k(x+y+z)(x-z)^2 then k is equal to

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

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 |[a+x,y,z],[x,a+y,z],[x,y,a+z]|=a^2(a+x+y+z)

Prove: |(y+z, z, y),( z, z+x,x),( y, x,x+y)|=4\ x y z

prove that: |(y+z,z,y),(z,z+x,x),(y,x,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)

Prove that : Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]]=xyz(x-y)(y-z)(z-x)

Prove that |[x,x^(2),x^(4)],[y,y^(2),y^(4)],[z,z^(2),z^(4)]|=xyz(x-y)(y-z)(z-x)(x+y+z)