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How did Arthur Cayley and James Joseph Sylvester develop the idea of matrices as mathematical objects?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In the 1850s, Arthur Cayley and his friend James Joseph Sylvester pioneered the concept of the matrix as an autonomous mathematical object. While the determinant was a primary influence, their work was also informed by earlier studies in number theory by Carl Friedrich Gauss and Ferdinand Gotthold Max Eisenstein.12

Sylvester is credited with introducing the term "matrix," while Cayley developed the formal algebraic aspects of these arrays in two influential papers. Cayley demonstrated that matrices could be manipulated similarly to ordinary numbers, subject to specific rules of arithmetic. Crucially, he recognized that matrix multiplication is not generally commutative, marking a significant departure from traditional number systems.21

Cayley applied these objects to solve systems of linear equations by utilizing the inverse matrix, effectively representing the solution through the analogy $x = a^{- 1} b$ . This development was supported by the British tradition of symbolic algebra, which shifted the focus of mathematics toward the study of operations among abstract objects.12