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:

(AB)T=BTAT

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]

Thank you!

**Answer**

One of my best college math professor always said:

Make a drawing first.

Although, he couldn’t have made this one on the blackboard.

**Attribution***Source : Link , Question Author : user1337 , Answer Author : mdup*