What is the significance of matrix multiplication

It is important to always think of a matrix as a representation of the transformed standard basis vectors rather than just thinking about a matrix as a rectangular array of random real numbers.

With this thought, matrix multiplication can be visualised as applying a series of transformations from right to left. For example, Lets say,

Then the matrix product AB is equivalent to first applying the transformation B and then applying the transformation A.

But on which vectors are we applying these transformations?

Well, its always the standard basis vectors ( and in this case) which are transformed (while keeping the origin constant). We can attach the matrix of standard basis vectors next to the matrix B to signify that, first the transformation B is applied on the standard basis vectors and, then the transformation A is applied on the resultant matrix.
Let S denote the 2×2 standard basis vector matrix,

Now the matrix multiplication can be written as, ABS

Starting from the right, lets first calculate the product BS.
We will find the transformations for the unit vectors and separately as vectors and then combine them into a matrix just like S. So, the transformation for the unit vector is,

and the transformation for the unit vector is,

Combining the above results in a matrix,

The next transformation matrix A is now applied on the product BS

Again, we will find the transformations for the unit vectors and separately as vectors and then combine them into a matrix. So, the transformation for the unit vector is,

and the transformation for the unit vector is,

Combining the above results in a matrix,