Why on earth is matrix multiplication NOT commutative ? – 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 :

AB \neq BA

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.

(Rotation)(Shearing) \neq (Shearing)(Rotation)

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

(Rotation)(Shearing)

2

(Shearing)(Rotation)

1

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.)

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