Discovering the theory of matrices: one of Arthur Cayley’s first papers


Matrices were discovered, or invented if you prefer to see it that way, by the British mathematician Arthur Cayley, who also did pioneering work in group theory and invariant theory.

This paper from 1857 (click here to read it) was not his very first. It was the one in which he announced the Cayley-Hamilton theorem, which says that, for every matrix A, if you plug in A instead of λ in the equation used to find eigenvalues of A, then the resulting equation in A is also true. He also explains the rules for multiplying matrices and summarises many other properties of matrices.

You’ll notice that here, as in all important writing about maths, there are quite a lot of words. Cayley does not “show all his working”. He makes his algebraic working as concise and compressed as possible. But he always uses words to explain what he is doing and the ideas behind the algebraic working.

The notation in the paper for matrices, with brackets round the first row connected to vertical lines around the other rows, looks odd to us today. I suspect it was dictated by what printers could manage in those days.


Cayley did much of his best maths in his spare time while also working in a day job as a lawyer. A condition for academic jobs at Cambridge University then was to go through the formal procedure of being ordained as a Church of England priest (though not having to do any priest-type work). Many people went through the formalities just for the sake of the job. Cayley refused on grounds of conscience. He trained as a lawyer in order to pay the rent while he was doing maths.

Later Cambridge University relaxed its foolish rules, and Cayley became professor of maths there in 1863. He supported the moves to allow women to study at Cambridge University. The first women came in 1869; the first woman to study maths, Sarah Woodhead, did her final exams in 1873. Despite Cayley’s efforts, though, Cambridge University would continue refusing to give degrees to women until 1948. Cayley was, apparently, a quiet, even-tempered man, easy to get on with, but not good at forceful campaigning.