Show that `a+b+c` is a factor of `|[a,b,c],[b,c,a],[c,a,b]|`

Show that a-b is a factor of |[1,a,a^(2)],[1,b,b^(2)],[1,c,c^(2)]|

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

Factorize the following: |[x+a,b,c], [b,x+c,a] , [c,a,x+b]|

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

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

Show without expanding at any stage that: [a+b,b+c,c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|=2|[a,b,c],[b,c,a],[c,a,b]|

If omega is a cube root of unity then |(a,b,c),(b,c,a),(c,a,b)| has a factor

Show that |[b+c,c+a,a+b] , [a+b,b+c,c+a] , [a,b,c]|=a^3+b^3+c^3-3abc

If a, b, c are positive and unequal, show that value of the determinant Delta=|(a,b,c),(b,c,a),(c,a,b)| is negative

Show that |[a,b,c],[a^2,b^2,c^2],[a^3,b^3,c^3]|=abc(a-b)(b-c)(c-a)