A proof of the Cayley-Hamilton theorem

Abstract

The standard proof to this theorem uses properties of the adjoint matrix combined with the use of the Leibnitz expansion whereby finally we get a number of simultaneous matrix equations which give us the required result. However, an alternative to this proof is the following structured proof that makes use of a lemma, a proposition and a theorem, finally giving us a neat proof of the actual theorem. In the process, several properties of linear transformations are highlighted and it is these properties which we will use to finally prove this theorem. This theorem 'will be divided into four parts.