Introduction

The Long History Behind a Short Word

The beginnings of matrices and determinants go back to the second century BC, with traces as far back as the fourth century BC — though it was only near the end of the seventeenth century that these ideas resurfaced and their real development began. It is no accident that matrices and determinants emerged from the study of systems of linear equations: the Babylonians tackled problems that led to simultaneous linear equations, and some of their work survives today on clay tablets.

The theory of matrices grew out of the search for compact, systematic methods of solving systems of linear equations, and also out of the study of transformations of geometric objects. In 1850, the English mathematician and lawyer James Joseph Sylvester coined the word 'Matrix' — from the Latin mater, meaning mother, the array from which further arrays (such as determinants) are "born."

The word determinant was first used by Carl Friedrich Gauss in 1801, in his Disquisitiones Arithmeticae, while he was studying quadratic forms — though not quite in today's sense; in the same work, he arranged the coefficients of his quadratic forms into rectangular arrays and, in doing so, described matrix multiplication. It was Augustin-Louis Cauchy (1812) who used determinant in its modern sense and studied it in depth, reproving earlier results and adding new ones on minors and adjoints. Arthur Cayley published a theory of determinants in 1841 — introducing the two-vertical-lines notation ($|\cdot|$) that is still standard today — and then, in an 1858 paper, gave the first fully abstract definition of a matrix, showing that the coefficient arrays already used for quadratic forms and linear transformations were special cases of his more general idea. Sylvester, William Rowan Hamilton (1805–1865), and Cayley are the three mathematicians most credited with shaping matrix theory; the now-familiar square-bracket notation for a matrix was introduced later still, by the English mathematician Cullis, in 1913.

Matrices matter well beyond mathematics itself — in genetics, economics, sociology, psychology, and industrial management — and specifically:

• as the coefficients of a system of linear equations,
• in spreadsheet-style tables of data (budgeting, sales projections, cost estimates, experimental results),
• to represent a geometric transformation (magnification, rotation, reflection), and
• in the structure of a network (social accounting and input-output tables in economics, communication/network analysis in electrical engineering, and cryptography).