resent any other n-by-nmatrix. We then have AIAand IAA. This is much
like the situation when using the real numbers: x 1 xand 1 xx.How are matrices used?
Matrices are used in a multitude of fields, from mathematics and science to certain
humanities fields. For example, they are used in physics to determine the equilibrium
of rigid bodies; in graph theory, fractals, and solutions of systems of linear equations
in mathematics; and in forest management, computer graphics, cryptology—even
electrical networks.ABSTRACT ALGEBRA
What is abstract algebra?
Abstract algebra is a collection of mathematical topics that deal with algebraic struc-
tures rather than the usual number systems. These structures include groups, rings,
and fields; branches of these topics include commutative and homological algebras. In
addition, linear algebra and even elementary number theory (see “Math Basics”) are
often included under abstract algebra.What is an algebraic structure?
An algebraic structure is made up of a set (collection of objects called elements; for
more information about sets, see “Foundations of Mathematics”) together with one or
more operations on the set that satisfy certain axioms. The algebraic structures get
their names depending on the operations and axioms. For example, algebraic struc-
tures include fields, groups, and rings, as well as many other structures with strange
names such as loops, monoids, groupoids, semigroups, and quasigroups.What is a field?
A field is an algebraic structure that shares the common rules for operations (addi-
tion, subtraction, multiplication, and division, except division by zero) of the rational,
real, and complex numbers (but not integers, see below under “ring”). A field must
have two operations, must have at least two elements, and must be commutative, dis-
tributive, and associative (see above for definitions). Formerly called “rational
domain,” a field in both French (corps) and German (Körper) appropriately means
“body.” A field with a finite number of members is called a Galoisor finite field. Fields
160 are useful to define such concepts as vectors and matrices.