The fact that Q^-1AQ=A satisfies means that AQ=QA. A could therefore be the identity matrix I_n. (The matrix I is called an identity matrix because IA = A and AI = A for all matrices A.)

Another possibility is that A is the inverse of Q. This is based on the definition of “inverse of a matrix”:

Definition of The Inverse of a Matrix

Let A be a square matrix of order n x n. If there exists a matrix B such that 
A B = I _n = B A 
Then B is called the inverse matrix of A. (and of course matrix A is the inverse matrix of B). 

There may be some other types of n x n matrices A that satisfy Q^-1AQ=A, but these are two possibilities that we were able to come up with immediately.