Connection between Leibniz formula and geometric interpretation of determinants

Question

I've seen that the Leibniz formula of determinants can be reduced to the Laplace expansion through consolidating terms. What I haven't been able to find online is a connection between any of those and the geometric interpretation of determinants, which is the volume (or measure) contained between the parallelopiped formed from the column (or row) vectors. How do I prove that the Leibniz formula, which is based on terms composed of permutations has anything to do with geometric measures?

Answer 1

As a function from $n \times n$ matrices to scalars, the determinant is uniquely determined by the following properties:

• It is multilinear as a function of the columns.
• It is alternating, meaning that it vanishes if two of the columns are the same, or equivalently (over a field of characteristic $\neq 2$) that it changes sign if you swap any two columns.
• $\det(I) = 1$.

This is a nice exercise. "Column" can also be replaced with "row" throughout. Once you've proven this, the argument proceeds in two phases:

1. First, show that over $\mathbb{R}$, the signed volume definition of the determinant satisfies all of the above properties.
2. Second, show either that the above properties imply the Leibniz formula, or that the Leibniz formula satisfies the above properties.