What is the essence of determinant?

[wdlang]

the definition of a determinant is not simple at the first sight

however we still have extremely simple relation det(AB)=det(A)det(B)

it seems that the grassmann algebra is the essence behind determinant, isn't it?

 

[Bacle]

Part of the problem is that a matrix can represent many different (and intrinsically different) things: a system of linear equations; a linear transformatio; the adjacency
relation in a graph, etc. , so I don't think a single interpretation for det is reasonable.

Do you have a particular use/context for a matrix in mind?