Without expanding the determinant , prove that `|{:(ax,by,cz),(x^2,y^2,z^2),(1,1,1):}|=|{:(a,b,c),(x,y,z),(yz,zx,xy):}|`

Prove that |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,b,,c),(x,,y,,z),(yz,,xz,,xy):}|

Prove that |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,b,,c),(x,,y,,z),(yz,,xz,,xy):}|

Prove that |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,c,,c),(x,,y,,z),(yz,,xz,,xy):}|

Prove that |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,c,,c),(x,,y,,z),(yz,,xz,,xy):}|

Prove that: |[ax,by,cz], [x^2,y^2,z^2], [1,1,1]| = |[a,b,c],[x,y,z],[yz,zx,yx]|

Without expanding the determinant, prove that (i) |{:(a,a^(2),bc),(b,b^(2),ca),(c,c^(2),ab):}|=|{:(1,a^(2),a^(3)),(1,b^(2),b^(3)),(1,c^(2),c^(3)):}| (ii) |{:(ax,by,cz),(x^(2),y^(2),z^(2)),(1,1,1):}|=|{:(a,b,c),(x,y,z),(yz,zx,xy):}| (iii) |{:(1,bc,b+c),(1,ca,c+a),(1,ab,a+b):}|=|{:(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2)):}|

Prove that : |{:(1,x,yz),(1,y,zx),(1,z,xy):}|=(x-y)(y-z)(z-x)

Show that Delta =Delta_1 where Delta=|{:(Ax,x^2,1),(By,y^2,1),(Cz,z^2,1):}|,Delta_1=|{:(A,B,C),(x,y,z),(zy,zx,xy):}|