Getting Down with Determinants: Defining det ( A ) Via the PA = LU Decomposition. Henry Ricardo Medgar Evers College (CUNY) Brooklyn, NY [email protected] Joint Mathematics Meetings San Diego, January 7, 2008. Seki and Leibniz. Cauchy. Cayley and Sylvester.
Getting Down with Determinants:Defining det (A)Via thePA = LU Decomposition
Medgar Evers College (CUNY)
Joint Mathematics Meetings
San Diego, January 7, 2008
…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,
whilethe role of determinants has been relegated to a minor backwater position.
—Carl D. Meyer
“Determinants are difficult, nonintuitive, and
often defined without motivation.”
“Down with Determinants!”
(Amer. Math. Monthly102 (1995), 139-154)
“It is hard to know what to say about
“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, butalso to decide
on its definition.”[emphasis mine]
- Gilbert Strang, Linear Algebra and Its Applications (3rdEdn.)
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.
Note: det (A) = (-1)(-1)(1)(-27) = -27
So by forward substitution (top-down)
So by back substitution c =
Standard properties of the determinant follow
easily from previous work with elementary
matrices and from the definition itself.