Did you know . . .?

The history of matrices goes back to ancient times! But the term "matrix" was not applied to the concept until 1850.

"Matrix" is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced.

The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of the use of matrix methods to solve simultaneous equations.

In the treatise's seventh chapter, "Too much and not enough," the concept of a determinant first appears, nearly two millennia before its supposed invention by the Japanese mathematician Seki Kowa in 1683 or his German contemporary Gottfried Leibnitz (who is also credited with the invention of differential calculus, separately from but simultaneously with Isaac Newton).

More uses of matrix-like arrangements of numbers appear in chapter eight, "Methods of rectangular arrays," in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by Carl Friedrich Gauss (1777-1855), also known as Gaussian elimination.

The term "matrix" for such arrangements was introduced in 1850 by James Joseph Sylvester.

Sylvester, incidentally, had a (very) brief career at the University of Virginia, which came to an abrupt end after an enraged Sylvester hit a newspaper-reading student with a sword stick and fled the country, believing he had killed the student!

James Joseph Sylvester.

Since their first appearance in ancient China, matrices have remained important mathematical tools. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps!

The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory.

In their own words:

"I did not look for matrix theory. It somehow looked for me." --Olga Taussky Todd in American Mathematical Monthly

(quotation source: http://www.maa.org/mathland/mathtrek_8_16_99.html)

"Mathematics is more than an art form. "--Sei Kowa

(quotation source: http://www-history.mcs.st-and.ac.uk/history/Quotations/Seki.html)

"Mathematics is the queen of the sciences and number theory is the queen of mathematics." --Carl Friedrich Gauss

"God does arithmetic." --Carl Friedrich Gauss

(source for Gauss quotations: http://www-history.mcs.st-and.ac.uk/history/Quotations/Gauss.html)

"...there is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being.... "--James Sylvester, Presidential Address to British Association, 1869.

"Mathematics is the music of reason."--James Sylvester

(source for Sylvester quotations: http://www-history.mcs.st-and.ac.uk/history/Quotations/Sylvester.html)

Applet to perform basic arithmetic operations on matrices: http://www.quickmath.com/www02/pages/modules/matrices/arithmetic/basic/index.shtml

Applet to calculate the inverse of a matrix: http://www.quickmath.com/www02/pages/modules/matrices/inverse/basic/index.shtml

Applet to calculate the determinant of a matrix: http://www.quickmath.com/www02/pages/modules/matrices/determinant/basic/index.shtml

Matrices and relationships: http://www.cut-the-knot.com/blue/relation.html

Contra-dancing and matrices: http://www.sciencenews.org/sn_arc97/6_14_97/mathland.htm

http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html#86

Matrix algebra tutorial: http://www.sosmath.com/matrix/matrix.html

On Olga Taussky Todd: http://www.maa.org/mathland/mathtrek_8_16_99.html

On Johann Carl Friedrich Gauss: http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Gauss.html

On James Sylvester: http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Sylvester.html

On Nine Chapters of the Mathematical Art: http://saxakali.com/COLOR_ASP/developcm3.htm

Multiplying matrices: http://www.math.toronto.edu/mathnet/plain/questionCorner/matrixmul.html

More uses of matrices:http://www.maa.org/mathland/mathtrek_2_1_99.html

By Laura Smoller, UALR Department of History.

April 2001.