Question

Show that it is not always true that AB = A^T B^T for matrices A, B.

Answer 1

To disprove something, you merely need to provide a counterexample.
Pick any two arbitrary matrices, of any size.
I picked
[1 2] and [1 3]
[0 1] and [0 1]

Transposing them you get
[1 0] and [1 0]
[2 1] [3 1]

Do the math yourself to verify it; you should get
[1 5] vs [1 0]
[0 1] [4 1]

Clearly they are not equal.