determianant help

[pxy2d1]

the problem is :

assuming that det(AB) = det(A)det(B) for 3x3 matrices prove that:

| 3 a+b+c a^3 + B^3 + c^3 |
|a+b+c a^2 + b^2 + c^2 a^4 + b^4 + c^4 |
|a^2 + b^2 + c^2 a^3 + B^3 + c^3 a^5 + B^5 + c^5 |

is equal to (a+b+c)(a-b)^2(a-c)^2(b-c)^2

im not sure im going about this problem in the right way. if i expand across the top row i seem to get the determinant to be equal to zero. anybody know if this is correct?
if so how can it then be equal to (a+b+c)(a-b)^2(a-c)^2(b-c)^2
would appreciate any help or ideas.
thanks

 

[pka]

The idea is to see that the given matrix can be written as a product:
\(\displaystyle \left| {\begin{array}{rrr}
1 & 1 & 1 \\
a & b & c \\
{a^3 } & {b^3 } & {c^3 } \\
\end{array}} \right|\left| {\begin{array}{rrr}
1 & a & {a^2 } \\
1 & b & {b^2 } \\
1 & c & {c^2 } \\
\end{array}} \right|.\)

 

[pxy2d1]

ah ok i see now.
if i now find the determinant of each of the matrices seperatley that make up matrix AB the determinant should be the same as that of matrix AB.
is there a way to find the two matrices systematically or have you simply looked at matric AB and deduced what the other two matrices are
thanks for your help

 

[pka]

is there a way to find the two matrices systematically or have you simply looked at matric AB and deduced what the other two matrices are

Years and years of experience.

 

[pxy2d1]

ok.
well thanks alot anyway