I'm not too good with matrix equations and how to go about solving or even just simplifying them. An example from my text book is:

"Suppose that A is a square matrix of size n x n that satisfies the the matrix equation

A^{15}A^{T}A^{14}+ 72A^{6}(A^{T})^{2}A^{5}+ 16AA^{T}A^{3}- 7I = 0

Show that the matrix A is invertible and find an expression for A^{-1}in terms of A"

How can I go about solving this? I don't even know where to start :S

Therefore, any help would be greatly appreciated.

Thanks,

Mark