# In what ways could Leibniz's formula of the determinant provide a deeper intuition?

Leibniz formula for the determinant

$$\det(A) =\sum_{\sigma \in S_n}\text{sgn}(\sigma )\prod_{i=1}^n a_{\sigma(i),i}$$

connects the determinant equation with group theory concepts, such as symmetrical groups and permutation parity. It's explanation has been covered here.

Computationally, it is proportional to $n!$ operations, making it impractical. There is definitely its aesthetic beauty, but does it provide a deeper intuition of the concept of determinant geometrically, or otherwise?

• This formula is obtained nearly immediately from the determinant as volume function, so it is not just aesthetic, it is very natural. It can also be used quite easily to deduce properties of the determinant. Why would you expect it to deepen one’s intuition though? – Ittay Weiss Nov 28 '17 at 18:02
• @IttayWeiss While Eigen concepts have clear intuitions in mechanical vectorial applications or in PCA analysis of variance, determinants (to the non-mathematician), and except in the context of parallelograms and parallelepipeds, are less intuitive. It is in this regard, that the connection to (signed) permutations and groups seems to open the door to a deeper understanding. – Antoni Parellada Nov 28 '17 at 18:09