Prove A^-1=(A^T A)^-1 A^T (matrices)

[blueyellow]

For a general, non-singular matrix A prove that

A^-1=[(A^T A)^-1] A^T

The Attempt at a Solution

tried searching in textbook and internet-nothing yet
someone somewhere must know an easy way to do this without having to sit there for five hours getting stuck
but i just look at this and i dnt even kno how to start
thanks in advance

 

[Berko]

The inverse of a product is the product of the inverses in reverse order.

 

[I like Serena]

$$((a^t a)^{-1}) a^t = ((a^t a)^{-1}) a^t ((a a^{-1}) = ((a^t a)^{-1}) (a^t a) a^{-1} = i a^{-1} = a^{-1}$$

[EDIT]For some unknown reason the forum translated A, T and I into lowercase letters.
Well, I hope you catch the drift.[/EDIT]