Determinant

Beta · **Joined:** Thu, 30 Oct 2008 10:47:09 UTC **Posts:** 182

Compute the determinant of the symmetric matrix
$\left(\begin{array}{ccccc} a_1b_1 & a_1b_2 & a_1b_3 & \cdots & a_1b_n\\ a_1b_2 & a_2b_2 & a_2b_3 & \cdots & a_2b_n\\ a_1b_3 & a_2b_3 & a_3b_3 & \cdots & a_3b_n\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1b_n & a_2b_n & a_3b_n & \cdots & a_nb_n\\ \end{array}\right)$

Any hint ?

robbwrr · **Posted:** Thu, 12 Mar 2009 17:18:22 UTC

$D=\left|\begin{array}{ccccc} a_1b_1 & a_1b_2 & a_1b_3 & \cdots & a_1b_n\\ a_1b_2 & a_2b_2 & a_2b_3 & \cdots & a_2b_n\\ a_1b_3 & a_2b_3 & a_3b_3 & \cdots & a_3b_n\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1b_n & a_2b_n & a_3b_n & \cdots & a_nb_n\\ \end{array}\right|$ $=a_1b_n \left|\begin{array}{ccccc} b_1 & b_2 & b_3 & \cdots & b_n\\ a_1b_2 & a_2b_2 & a_2b_3 & \cdots & a_2b_n\\ a_1b_3 & a_2b_3 & a_3b_3 & \cdots & a_3b_n\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1 & a_2 & a_3 & \cdots & a_n\\ \end{array}\right|$

Now subtract $b_k \cdot R_n$ (bottom row) from R_k

(k-th row) for each k>1.
Expand that by minors (choose the row with all zeros but one) to get:

Spoiler:

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Determinant

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