Find the value of determinant `|[a+b,b,c],[b+c,c,a],[c+a,a,b]|`

Let a ,b ,c be positive and not all equal. Show that the value of the determinant |[a, b, c],[b, c, a],[ c, a ,b]| is negative.

If in a Delta ABC, sin ^(3) A + sin ^(3) B+ sin ^(3) C =3 sin A .Sin B. sin C, then find the valueof determinant |{:(a,b,c),(b,c,a),(c,a,b):}|.

Write the value of the determinant |[a,1,b+c], [b,1,c+a] ,[c,1,a+b]| .

The value of the determinant |[a-b,b+c,a],[b-c,c+a,b],[c-a,a+b,c]| is

If a ne b ne c are all positive, then the value of the determinant |{:(a,b,c),(b,c,a),(c,a,b):}| is

Without expanding the determinant, show that (a+b+c) is a factor of the determinant |[a, b, c], [b, c, a], [c, a, b]|

The value of the determinant |[a-b,b+c, a], [b-a, c+a, b], [c-a, a+b, c]| is (a) a^3+b^3+c^3 (b) 3\ b c (c) a^3+b^3+c^3-3a b c (d) none

Find the value of the determinant |{:(a^2,a b, a c),( a b,b^2,b c), (a c, b c,c^2):}|

Prove the following, using properties of determinants: |[a+b+2c,a,b],[c,b+c+2a,b],[c,a,c+a+2b]|=2(a+b+c)^3

The value of the determinant |(b +c,a -b,a),(c +a,b -c,b),(a +b,c -a,c)| , is