The eigenvalues (and so the characteristic polynomial) determine a matrix up to "similarity".
Matrices having the same eigenvalues (characteristic polynomial) must be "similar"- that is, for some invertible matrix Q.
I am wondering whether it is possible to determine a matrix if its characteristic polynomial is known. Of course, there is more than one matrix for a fixed characteristic polynomial but there must also be something special that other matrixes dont have. For instance, if the c.p. is with n the order, are there some simple conditions that can be applied to its matrix?