How can two matrices cancel each other out when there is a matrix in between?

Question

I'm watching videos of Gilbert Strang's linear algebra lectures. In lecture 17, where he goes over orthonormal bases and the Gram-Schmidt process, he proves

$$ A^TB = A^T\left( b - \frac{A^Tb} {(A^TA)} A\right) = 0 $$

by having

$$ A^T \frac{A^Tb} {(A^TA)} A $$ cancel out into A^Tb. I don't know how the A^TA on the top was allowed to cancel out with the A^TA on the bottom if there is an A^Tb sandwiched in between the A^T and the A on the numerator. Shouldn't the A^T and the A on the numerator not be allowed to multiply each other?

I wondered if the A^T on the numerator can cancel out with the A^T on the denominator and the same for the two A's, but I do not know if this violates the order of operations for matrix multiplication. If I were to write (A^TA)^-1 rather than have the (A^TA) on the denominator below the (A^Tb), where would it go?

Answer 1

Here's Prof. Strang's blackboard at 33m56s:

Note that $\mathrm A, \mathrm B$ are vectors, not matrices!! In other words, the question makes no sense whatsoever.