Although 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 purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, 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.”

The ancient Mayans may have been among the first cultures to use a simple form of matrices.

But the matrix story was not all about Sylvester. In 1845, Sylvester’s collaborator, English mathematician Arthur Cayley (1821–1895), used a form of matrices 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 American Journal of Mathematics in 1878; and at the ripe age of 71, he invented the theory of reciprocants (differential invariants).