summation of determinants of $3\times3$ matrices I have an algebra problem but no idea how to solve it. The problem is: "you can create 9! matrices the elements of which lie in a set $ \{1,2,3,...,9\} \subset \mathbb N$ so that their elements do not repeat, i.e. e.g.
$$
\begin{pmatrix}1&2&9\\3&5&7\\6&4&8 \end{pmatrix}
$$
Find the sum of the determinants of all these matrices."
Could you give me a hint how to solve it? Thank you.

The number of those matrices is even, and note that the sum of the determinants of the pair $\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}$ and $\begin{pmatrix}b&a &c\\e&d&f\\ h&g& i\end{pmatrix}$ is 0. If we pair up these matrices in this way we can see that the required sum is 0.
