Without expanding show that following : `|[a,a+b,a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b,10a+6b+3c]| = a^3`

Prove that |(a,a+b, a+b+c),(2a,3a+2b,4a+3b+2c),(3a,6a+3b,10a,6b+3c)| = a^3

Expand the following (3a-2b)^3

Without expanding the following determinant, show that : |[3a+b, 2a, a],[4a+3b, 3a, 3a],[5a+6b,4a,6a]|=a^3

Without expanding, prove the following |(0,ab^2,ac^2),(a^2b,0,bc^2),(a^2b,b^2c,0)|=2a^3b^3c^3

Without expanding, prove the following |(a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b)|=a^3+b^3+c^3-3abc

Expand the following (3a-5b-c)^2

Without expanding show that the following determinants vanish |(4,a,b+c),(4,b,c+a),(4,c,a+b)|

Without expanding the determinant, prove that |[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]|