Getting Down with Determinants: Defining det (A) Via the PA = LU Decomposition Henry Ricardo Medgar Evers College (CUNY) Brooklyn, NY Joint Mathematics Meetings San Diego, January 7, 2008

Seki and Leibniz

Cauchy

Cayley and Sylvester

…mathematics, like a river, is everchanging in its course, and major branches can dry up to become minor tributaries while small trickling brooks can develop into raging torrents. This is precisely what occurred with determinants and matrices. The study and use of determinants eventually gave way to Cayley’s matrix algebra, and today matrix and linear algebra are in the main stream of applied mathematics, while the role of determinants has been relegated to a minor backwater position. —Carl D. Meyer

“Determinants are difficult, nonintuitive, and often defined without motivation.” —Sheldon Axler “Down with Determinants!” —Sheldon Axler (Amer. Math. Monthly 102 (1995), )

“It is hard to know what to say about determinants….” “There is one more problem about the determinant. It is difficult not only to decide on its importance, and its proper place in the theory of linear algebra, but also to decide on its definition.” [emphasis mine] - Gilbert Strang, Linear Algebra and Its Applications (3 rd Edn.)

The Usual Suspects Laplace’s expansion (minors/cofactors) Permutation definition Unique alternating multilinear function Signed volume of a parallelepiped

Preliminaries I Gaussian elimination –elementary row operations –REF (not unique) –RREF (unique) Elementary matrices (3 types) –elementary row ops via premultiplication Invertible matrices

Preliminaries II LU Decomposition –doesn’t always exist –is unique if a nonsingular matrix has an LU factorization –can be used to solve a system:

Preliminaries III PA = LU Decomposition –always exists –is unique if A is nonsingular (i.e., prod. of is unique) –can be used to solve a system:

DEFINITION Given an n × n matrix A, the determinant of A, denoted det (A) or |A|, is defined as follows: where and k is the number of row interchanges represented by P.

Example

Example (Cont.) Note: det (A) = (-1)(-1)(1)(-27) = -27. So by forward substitution (top-down)

Example (Cont.) So by back substitution c =

Properties Standard properties of the determinant follow easily from previous work with elementary matrices and from the definition itself.

Summary The determinant appears as a natural consequence of using the PA = LU factorization to solve a system of equations. “Handwaving” is minimal. Standard properties of an n x n determinant follow easily.