Hi MHF,
I'm having heaps of trouble working out how to solve this question. The text book that I am using is the third addition of Lay's Linear Algebra book and I cant find any examples on this type of question..
Suppose A is a square matrix which satisfies the matrix equation
Determine whether A is invertible or not. If it is invertible, then provide an expression for the inverse of A.
Be careful, this is not necessarily the characteristic polynomial. The characteristic polynomial can be writen as (Cayley-Hamilton) where is minimal polynomial for . Since solves for it means must divide this polynomial. But this polynomial is irreducible by Eisenstein criterion. Thus, the characteristic polynomial for must be of form . Now we see the constant term is non-zero and so .