Prove the determinant inequation.

Question

Show that $\begin{vmatrix} a^2 & 2ab & b^2\\ ac & ad+bc &bd \\ c^2& 2cd & d^2 \end{vmatrix}\geq 0$, where $a$, $b$, $c$ and $d$ are real numbers.

Answer 1

I don't buy it. The determinant is equal to $(ad-bc)^3$, from which it follows that if $ad-bc<0$, the assertion is false.

Answer 2

this is Important for fuchsian groups.

$$ H = \left( \begin{array}{rrr} 0 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 0 \end{array} \right) $$

Given $$ Q = \left( \begin{array}{ccc} \alpha^2 & 2 \alpha \beta & \beta^2 \\ \alpha \gamma & \alpha \delta + \beta \gamma & \beta \delta \\ \gamma^2 & 2 \gamma \delta & \delta^2 \end{array} \right), $$ there is a typographical error on page 23 of Magnus (Noneuclidean Tesselations and Their Groups), actually $\det Q = (\alpha \delta - \beta \gamma)^3.$ We have the identity $$ Q^T H Q = (\alpha \delta - \beta \gamma)^2 H. $$