Determinant

rebirthflame · #1 $Old$ December 1st, 2008, 03:15 AM

Assuming Det(AB) = Det(A) x Det(B) for 3x3 matrices prove that the determinant of

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

= (a+b+c)(b-a)^2(c-a)^2(a-b)^2

i was just going to work through the math but i dont really understand how the result given at the top will help me when working out the determinant as its is the cofactor times the 2x2 you get - the next cofactor times the 2x2 and so on.

flyingsquirrel · #2 $Old$ December 1st, 2008, 07:52 AM

Hi,

Quote:

Originally Posted by rebirthflame $View Post$

Assuming Det(AB) = Det(A) x Det(B) for 3x3 matrices prove that [...]

$\begin{vmatrix}3 & a+b+c & a^3+b^3+c^3\\a+b+c & a^2+b^2+c^2 & a^4+b^4+c^4\\ a^2+b^2+c^2 & a^3+b^3+c^3 & a^5+b^5+c^5\\\end{vmatrix}= (a+b+c)(b-a)^2(c-a)^2(a-b)^2$

I suggest you try to find two matrices $A$ and $B$ such that

$AB=\begin{pmatrix}3 & a+b+c & a^3+b^3+c^3\\a+b+c & a^2+b^2+c^2 & a^4+b^4+c^4\\ a^2+b^2+c^2 & a^3+b^3+c^3 & a^5+b^5+c^5\\\end{pmatrix}$