`|[b^2c^2,bc,b+c] , [c^2a^2,ca,c+a] , [a^2b^2,ab,a+b]|=0`

|[b^(2)c^(2),bc,a-c],[c^(2)a^(2),ca,b-c],[a^(2)b^(2),ab,0]|=?

Evaluate |[0,c,b] , [c,0,a] , [b,a,0]| hence show that |[0,c,b] , [c,0,a] , [b,a,0]|^2= |[b^2+c^2,ab,ac] , [ab,c^2+a^2,bc] , [ca,bc,a^2+b^2]|=4a^2b^2c^2

If a,b,c are non-zero real numbers then D=det[[b^(2)c^(2),bc,b+cc^(2)a^(2),ca,c+aa^(2)b^(2),ab,a+b]]=(A)abc(B)a^(2)b^(2)c^(2)(C)bc+ca+ab(D)0,

Without expending, prove that : (i) |{:(b^(2)c^(2),bc,b+c),(c^(2)a^(2),ca,c+a),(a^(2)b^(2),ab,a+b):}|=0 (ii) |{:(x,y,z),(x^(2),y^(2),z^(2)),(yz,zx,xy):}|=|{:(1,1,1),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^(3)):}| (iii) |{:(1,2x,x^(2)-yz),(1,y,y^(2)-zx),(1,z,z^(2)-xy):}| ("Taking 2, 3 and "2/3"common from "C_(1),C_(2)" and "C_(3)" repectively") =4xx49 ["from eq.(1)"] =198. (iv) |{:(sinx,cosx,sin(x+alpha)),(siny,cosy,sin(y+alpha)),(sinz,cosz,sin(z+alpha)):}|=0

Show that |[0,c,b] , [c,0,a] , [b,a,0]|^2=|[b^2+c^2, ab, ac] , [ab, c^2+a^2, bc] , [ac, bc, a^2+b^2]|

Prove that : |{:(b^(2)c^(2),bc, b+c),(c^(2)a^(2),ca, c+a),(a^(2)b^(2),ab, a+b):}|=0

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Prove that |[1,1,1] , [a,b,c] ,[a^2-bc, b^2-ca, c^2-ab]|=0