What is Cayley–Hamilton theorem? The Cayley–Hamilton theorem states that substituting the matrix A for x in polynomial, p(x) = det(xI n – A) , results in the zero matrices, such as: p(A) = 0 It states that a ‘n x n’ matrix A is demolished by its characteristic polynomial det(tI – A), which is monic polynomial of degree n. The powers of A, found by substitution from powers of x, are defined by recurrent matrix multiplication; the constant term of p(x) provides a multiple of the power A 0 , where power is described as the identity matrix. The theorem allows A n to be articulated as a linear combination of the lower matrix powers of A. If the ring is a field, the Cayley–Hamilton theorem is equal to the declaration that the smallest polynomial of a square matrix divided by its characteristic polynomial. Example of Cayley-Hamilton Theorem 1.) 1 x 1 Matrices For 1 x 1 matrix A(a 1,1 ) the characteristic polynomial is given by � ( � ) = � – � So, p(A) = (a) – (a 1,1 ) = 0 is obvious. 2