Determinant notes in PDF form.
The notes are intended for first year Math 1410 students although they could also be used for Math 3410. The approach is similar to that presented by Kenneth P. Bogart in A Fresh(man) Treatment of Determinants, Amer. Math. Mon., 96 (1989), 915-920. See also Linear Algebra. A First Course, with Applications to Differential Equations, Tom M. Apostol, John Wiley & Sons Inc, 1997.
Determinants are motived geometrically and a simple set of axioms based on row operations are used. From the axioms it is not hard to derive the key properties of determinants and to show the Laplace expansion formula. Using the latter the determinant is shown to exist and to be uniquely defined. The notes prove most of the well known properties for determinants. Determinants are used in the substitution formula for multiple integrals, and the geometric approach taken has obvious utility there.
It seems many textbook authors prefer to define determinants starting with Laplace expansion instead. While this approach may be acceptable if you are quickly passing by on your way to eigenvalues and eigenvectors, it is not so good in most other respects. The definition is very hard to motivate geometrically or any other way. Deriving the homomorphism property det(AB)=det(A)det(B) from it is messy. Laplace expansion cannot be used to compute a numerical determinant for any but the smallest orders either.
Last update: 2006 November 7