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?
Best 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.