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. Learn to state the Cayley-Hamilton Theorem, its proof, 3x3 matrix cases, applications, inverse method, and solved examples in linear algebra. In linear algebra, the Cayley–Hamilton theorem (termed after the mathematicians Arthur Cayley and William Rowan Hamilton) says that every square matrix over a commutative ring (for instance the real or complex field) satisfies its own characteristic equation. 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.
