Why on earth is matrix multiplication NOT commutative ? – An Intuition

One is commonly asked to prove in college as part of a linear algebra problem set that matrix multiplication is not commutative. i.e If A and B are two matrices then :

But without getting into the Algebra part of it, why should this even be true ? Let’s use linear transformations to get a feel for it.

If A and B are two Linear Transformations namely Rotation and Shear. Then it means that.

Is that true? Well, lets perform these linear operations on a unit square and find out:



You can clearly see that the resultant shape is not the same upon the two transformations.

This means that the order of matrix multiplication matters a lot ! ( or matrix multiplication is not commutative.)

Have a great day!


I used to work at the museum as the guy that told people not to touch the mummies. It was normally easy because most people are respectful or too stupid to realize that the mummies could be touched, but one old woman came in every year on the same day and made my job tough. She wore a shirt that said “#1 Mummy Toucher” and didn’t even look at the other exhibits, even though we had some cool ones like King Arthur’s round tablecloth and Kennedy’s novel about the Matrix. This woman would hop right over the velvet “do not hop over” rope and make a bee-line for the most touchable mummy. One year I was a step ahead of her and dressed up as that mummy, so when she went to touch me I grabbed her hand and said, “Don’t touch the mumm-ME!” and laughed and laughed. She didn’t think it was very funny though and had a big heart attack and died on the spot. I didn’t want to get in trouble so I threw her into one of the empty mummy sarcophagi. I think she’s still there today. Anyway, obviously I got promoted from that job to tour guide, so if you’ll follow me, we’ll check out Kennedy’s novel that caused the Wachowskis to kill him.

Basis Vectors are instructions!

Basis vectors are best thought of in the context of roads.

Imagine you are in a city - X which has only roads that are perpendicular to one another.

You can reach any part of the city but the only constraint is that you need to move along these perpendicular roads to get there.

Now lets say you go to another city-Y which has a different structure of roads.

In this case as well you can get from one part of the city to any other, but you have to travel these ‘Sheared cubic’ pathways to get there.

Just like these roads determine how you move about in the city,

Basis Vectors encode information on how you move about on a plane.

What do I mean by that ?

The basis vector of City-X is given as:

This to be read as - “ If you would like to move in City-X you can only do so by taking 1 step in the x-direction or 1 step in the y-direction ”

The basis vector of City-Y is given as:

This to be read as - “ If you would like to move in City-Y you can only do so by taking 1 step in the x-direction or  1 step along the diagonal OB ”


By having the knowledge about the Basis Vectors of any city, you can travel to any destination by merely scaling these basis vectors.

As an example, lets say need to get to the point (3,2), then in City-X,  you would take 2 steps in the x-direction and 3 steps in the y-direction

And similarly in City-Y, you would take 1 step along the x -direction and 2 steps along the diagonal OB.

Destination Arrived :D