linear algebra - A inverse times A in the middle of a matrix multiplication

I was wondering if for the below matrix multiplication:

$A^T * A *\ A^{-1} * (A^{-1})^T$

we can assume the product of the inner 2 matrices to equal the identity matrix I, and simply rewrite this as:

$A^T * (A^{-1})^T$

or is this not generally acceptable because matrix multiplication is not commutative?

Answer

You are correct. Since matrix multiplication is associative, thus you can do

$A^T \cdot A \cdot A^{-1} \cdot(A^{-1})^T = A^T \cdot (A \cdot A^{-1}) \cdot (A^{-1})^T = A^T \cdot (A^{-1})^T$

But you can even go further by switching the inverse with the transpose:

$A^T \cdot (A^{-1})^T = A^T \cdot (A^{T})^{-1} = I$

So your whole expression is equal to the identity matrix.