Engineering Mathematics, Volume II, Second Edition

is called the characteristic function of the matrix A. Concerning this function, we have the following famous theorem.

Theorem 2.62. (Cayley-Hamilton Theorem).

Every square matrix A satisfies its characteristic equation ϕ (A) = 0.

Proof. The characteristic matrix of A is A – λI_n. Since the elements of A – λI_n are at most of the first degree in λ, the elements, (cofactor) of the adjoint matrix of A – λI_n are of degree utmost n – 1 in λ. Therefore, we may represent adj (A – λI_n) as a matrix polynomial

 

 

where B_k is the matrix whose elements are the coefficients of λ^k in the corresponding elements of adj(A – λI_n). But

 

 

 

Substituting ...