TITLE: summation of determinants of $3\times3$ matrices
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: Given that others have answered, and the basic trick is the same, here is another way of thinking about the problem.
If I exchange the first and second rows of every matrix, I change the sign of every determinant and hence the sign of the sum. On the other hand, I get exactly the same set of matrices, so the sum must stay the same. This leaves me with only one possibility.
I mention it because it deals with the problem globally, and that kind of global reasoning is sometimes very useful. The local information is hidden in "I get exactly the same set of matrices" and if you write down the detail of what makes that obvious you will find yourself mirroring the other suggestions people have posted.