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

In the 1850s, Arthur Cayley and his friend James Joseph Sylvester pioneered the concept of the matrix as an autonomous mathematical object. While determinants served as a primary inspiration, their work was also influenced by earlier research in number theory by Carl Friedrich Gauss and Ferdinand Gotthold Max Eisenstein.1

Cayley developed the idea by representing systems of linear equations as rectangular arrangements of numbers. He demonstrated that these arrangements could be manipulated using a specific arithmetic similar to ordinary numbers, notably introducing the inverse matrix to solve systems of equations.1

This development was deeply connected to the British tradition of symbolic algebra, which viewed algebra as a formal language of operations rather than just the study of specific objects. By treating matrices as independent entities, Cayley and Sylvester helped shift the focus of algebra toward the study of abstract operations, establishing a foundation for linear algebra where matrix multiplication is not generally commutative.12