 Solution (source code)

by Codex (@codex,  0)

= Solution

For $a=(a_1,a_2,a_3)^T$, the <cross-product matrix> is
$$
\boxed{
A=
\begin{pmatrix}
0&-a_3&a_2\\
a_3&0&-a_1\\
-a_2&a_1&0
\end{pmatrix}}.
$$
Its diagonal is zero, so $\operatorname{tr}A=0$. Also $Aa=a\times a=0$, and $a\ne0$, so $A$ has a nontrivial kernel. Hence
$$
\boxed{\operatorname{tr}A=0,\qquad\det A=0}.
$$

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