What a Determinant Measures

Question

Compute the area of the parallelogram spanned by u = (3, 1) and v = (1, 2). Then explain, in one tight paragraph, what this has to do with determinants, with change of variables, and with knowing when two instruments are redundant.

Accepted Answer

Computation. detbeginpmatrix3 & 1 1 & 2endpmatrix = 3cdot2 - 1cdot1 = mathbf5, and the area is |det| = 5. Why determinant = area. The cleanest argument is axiomatic: area on parallelograms is linear in each spanning vector, equals 1 on the unit square, and vanishes when the vectors coincide, and those three properties uniquely pin down the determinant. Shearing one vector along the other (v mapsto v + tu) changes neither the area (same base, same height) nor the determinant (row operation), which is exactly why row reduction computes determinants. The sign tracks orientation: swap u and v and the determinant flips. The three quant payoffs. 1. Change of variables. A linear map M scales every area by |det M|, that's the Jacobian factor in every multivariate substitution. The r in r,dr,dtheta that unlocks the Gaussian integral is precisely the local determinant of the polar map; Monte Carlo importance-sampling weights are this idea repeated. 2. Redundancy detection. det = 0 means the spanned parallelogram is flat, the vectors are collinear. For two instruments with payoff/holdings vectors u and v, determinant zero says one is a multiple of the other: no new exposure, no way to imply t

What a Determinant Measures

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