I think the volume explanation is one of the most intuitive pieces of math in existence, personally! Uninvertibility of a tranformation corresponds to a volume of zero because the transformation must squish two dimensions together, leaving them impossible to differentiate, det(AB) = det(A)det(B) because applying two transformations applies their scaling successively, det(A^-1) = 1/det(A) because you have to undo the scaling to invert a transformation etc. I don't think the permutation definition is even strictly necessary; if I recall correctly Linear Algebra Done Wrong defines the determinant in terms of its geometric definition and develops its formula from the properties it must have. I think that the concept is well worth the investment of initial confusion. Axler disagrees, however.