1. ## 3x3 matricies proof

Assuming that det(AB) = det(A)det(B) for 3x3 matricies, prove that...

| a+b+c , a^2+b^2+c^2 , a^3+b^3+c^3|
|a^2+b^2+c^2 , a^3+b^3+c^3 , a^4+b^4+c^4| = abc(b-c)^2(c-a)^2(a-b)^2
|a^3+b^3+c^3 , a^4+b^4+c^4 , a^5+b^5+c^5|

The problem I encounter with this problem is that I either find that I get an answer of 0 or I end up with a very large equation which I can't find a way to cancel. Any help would be apreciated.

Rodregez

2. Multiply $A=\left(\begin{array}{ccc}1&1&1\\a&b&c\\a^{2}&b^{2 }&c^{2}\\\end{array}\right)$ and $B=\left(\begin{array}{ccc}a&a^{2}&a^{3}\\b&b^{2}&b ^{3}\\c&c^{2}&c^{3}\\\end{array}\right)$ and then use the given identity.

The determinants of $A$ and $B$ factor easily using elementary operations.

3. Originally Posted by BobP
Multiply $A=\left(\begin{array}{ccc}1&1&1\\a&b&c\\a^{2}&b^{2 }&c^{2}\\\end{array}\right)$ and $B=\left(\begin{array}{ccc}a&a^{2}&a^{3}\\b&b^{2}&b ^{3}\\c&c^{2}&c^{3}\\\end{array}\right)$ and then use the given identity.

The determinants of $A$ and $B$ factor easily using elementary operations.
Thank you, im still struggling with this question though, I cant get determinant B to come out right, i figure det(A) = (c-b)(b-a)(a-c) which looks promising but i get det(B) = (c-b)(bc-a(c+b)+a^2) so im not sure how to proceed from there really.

Thank you, Rodregez

4. ## Factors

For figuring out det(B), factor out an 'a' from the 1st line, a 'b' from the 2nd line and a 'c' from the 3rd. Then det(B) = abc det(new matrix). Note the new matrix is just A transposed.