It is well known that for invertible matrices A,B of the same size we have (AB)−1=B−1A−1
and a nice way for me to remember this is the following sentence:
The opposite of putting on socks and shoes is taking the shoes off, followed by taking the socks off.
Now, a similar law holds for the transpose, namely:
for matrices A,B such that the product AB is defined. My question is: is there any intuitive reason as to why the order of the factors is reversed in this case?
[Note that I’m aware of several proofs of this equality, and a proof is not what I’m after]
One of my best college math professor always said:
Make a drawing first.
Although, he couldn’t have made this one on the blackboard.