“Down with Determinants”

It’s always pleasant to find a pedagogical decision being supported by a higher authority. In my case, the decision was to eliminate determinants from an elementary linear algebra course, and the higher authority is Sheldon Axler from Michigan State University, whose magnificent article “Down with Determinants” (American Mathematical Monthly, 102, 1995, pp139-154); online at http://www.axler.net/DwD.html, won the Lester R. Ford award for the best expository mathematical article published by the Mathematical Association of America in its year.

Most approaches to linear algebra treat eigenvalues and eigenvectors as secondary to determinants: first you define the determinant of a matrix , and then you define the eigenvalues as the roots of the characteristic equation . Axler first defines eigenvalues of a linear transformation in a natural and simple manner: an eigenvalue is a value for which is not injective. In matrix language, this means that there is a non zero vector for which . His proof that every linear transformation has an eigenvalue is simple and elegant, and I fully agree with his comment that this proof “should be the one imprinted on our minds, written on our blackboards, and published in our textbooks.” Axler does define determinants, but instead of a messy computation involving matrix elements (come on, own up, how many of you have taught the “expansion by row or column” method for evaluating determinants?), he simply defines the determinant to be the product of all eigenvalues, counting multiplicities. This again follows naturally and simply from his previous discussion of “determinant-less” eigenvalues. And the characteristic equation and the Cayley-Hamilton theorem follow very easily.

Read this paper!

A nice supplement to this paper is one by Barton Willis from the University of Nebraska, on how to compute eigenvalues without determinants, called “Eigenvalues by row operations“.

My own teaching needs for linear algebra are meagre; in the first subject I teach we get up to matrix inversion and solution of linear equations, with a discussion of ranks. Since all of this can be done entirely with row operations, there is no need to invoke determinants at all, and the adjoint-determinant method for computing inverses need not be taught. As for Cramer’s rule, who in a first course needs it? I and my students have been much happier since I consigned determinants to the bin.