Arthur Cayley (1821 – 1895) was a British mathematician. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years. He published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work is in developing the algebra of matrices, work in non-euclidean geometry and n-dimensional geometry.
He was consequently able to prove the Cayley-Hamilton theorem -- that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way -- as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups.
In 1925, it was discovered that matrices and complex numbers were a necessary tool in describing the behaviour of atomic systems.