ent variables or numbers by addition, subtraction, multiplication, or other methods.
Each row and column of a given matrix must have the same number of elements.
Any time one has a list of numbers, or a table of numbers in a specific order, con-
cerning anything at all (prices, grades, populations, coordinates of points, production
tables ...), it can be considered to be a matrix. When the idea of the matrix was first con-
ceived, its development dealt with transformation of geometric objects and solution of
systems of linear equations. Historically, the early emphasis was on the determinant (see
below), not the matrix; today, especially in linear algebra, matrices are considered first.
What are some examplesof matrices?
The dimensions of a matrix are the number of rows (horizontal numbers) and
columns (vertical numbers); it is written as the rows first, columns second. The fol-
lowing are some simple examples of matrices, all of which differ in their dimensions: 157
ALGEBRA
Who invented matrices?
A
lthough a simple form of matrices may have been used by the Mayans (and
maybe other cultures; see below), the true mathematical use of a matrix was
first formulated around 1850 by English mathematician, poet, and musician
James Sylvester (1814–1897). In his 1850 paper, Sylvester wrote, “For this pur-
pose we must commence, not with a square, but with an oblong arrangement of
terms consisting, suppose, of mlines and ncolumns. This will not in itself rep-
resent a determinant, but is, as it were, a Matrix out of which we may form vari-
ous systems of determinants by fixing upon a number p,and selecting at will p
lines and pcolumns, the squares corresponding of pth order.” In this case,
Sylvester used the term matrix to describe its conventional use, or “the place
from which something else originates.”
But the matrix story was not all about Sylvester. In 1845 Sylvester’s collabo-
rator, English mathematician Arthur Cayley (1821–1895), used a form of matri-
ces in his work On the Theory of Linear Transformations;by 1855 and 1858,
Cayley began to use the term “matrix” in its modern mathematical sense.
Although he was an avid mountaineer and a lawyer for close to a decade and a
half (which is how he met Sylvester), during his free time Cayley published more
than 200 mathematical papers. He also contributed a great deal to the field of
algebra, initiated analytic geometry of n-dimensional spaces, and developed the
theory of invariants, among other mathematical feats.
Sylvester also remained brilliant throughout his life. He founded the Ameri-
can Journal of Mathematicsin 1878; and at the ripe age of 71, he invented the
theory of reciprocants (differential invariants).