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 :

image

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.

image

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

(Rotation)(Shearing)

(Shearing)(Rotation)

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!

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