Learn to state the Cayley-Hamilton Theorem, its proof, 3x3 matrix cases, applications, inverse method, and solved examples in linear algebra. The Cayley-Hamilton theorem is about the characteristic equation of a square matrix. Using this theorem one can find the inverse of a matrix, the integral power of a matrix, and many more. In this post, we will study this theorem along with some applications. The Cayley-Hamilton theorem produces an explicit polynomial relation satisfied by a given matrix. In particular, if M M is a matrix and p M (x) = det (M x I) pM (x) = det(M − xI) is its characteristic polynomial, the Cayley-Hamilton theorem states that p M (M) = 0 pM (M) = 0. The Cayley-Hamilton Theorem states that any square matrix satis es its own characteristic polynomial. The Jordan Normal Form Theorem provides a very simple form to which every square matrix is similar, a consequential result to which the Cayley-Hamilton Theorem is a corollary.
