What Vol I left unexplained
You can already compute a determinant: expand along a row, or reduce to triangular form and multiply the pivots. But why does swapping two columns flip the sign? Why is det of a product the product of dets? Vol I gave rules; it never said where they come from. The honest answer is that all of those rules follow from three simple properties — and any function with those three properties is forced to equal the determinant.
The shift in viewpoint is small but decisive. Stop thinking of a matrix A as a grid of numbers. Think of it as a LIST of its column vectors: A = [c_1, c_2, ..., c_n]. Then det becomes a function det(c_1, ..., c_n) of n vectors in R^n, returning a single scalar.
Three demands on the function
Here are the three properties. (1) Multilinear: det is linear in each column separately, holding the others fixed. (2) Alternating: if any two columns are equal, det = 0. (3) Normalized: det(I) = 1, i.e. det of the standard basis vectors is 1. That is the entire definition. Memorize these three lines; the rest of this track unpacks them.
A function that is linear in each of several vector arguments is called a multilinear map; one that vanishes on repeated arguments is an alternating multilinear form. So the slogan is: the determinant is the alternating multilinear form on columns, normalized so the identity scores 1.
Multilinearity, column j scaled and added: det(.., a*u + b*v, ..) = a*det(.., u, ..) + b*det(.., v, ..) Alternating, two equal columns: det(.., w, .., w, ..) = 0 Normalization: det(e_1, e_2, ..., e_n) = det(I) = 1 Quick check in 2x2 with [a c; b d] = [col1 col2]: det([a,b], [c,d]) and the three rules will force det = a*d - b*c (shown in Guide 2).
Why these three? A geometric preview
These rules are not arbitrary — they are exactly the rules of SIGNED VOLUME. Stretching one edge of a parallelepiped by factor a multiplies its volume by a (linearity). A box with two identical edges is flat, so it has zero volume (alternating). The unit cube has volume 1 (normalization). We pursue this fully in Guide 3 as the determinant as signed volume.
One immediate consequence is worth seeing now. If the columns are linearly dependent, det = 0. Reason: a dependent column is a linear combination of the others; expand by multilinearity into a sum of dets each having a repeated column, and every such term is zero by the alternating rule. So det detects linear independence — a flat box has no volume.