In the lecture on Matrix multiplication the Instructor showed how we combine two linear transformations as matrix multiplications. Here is the snapshot:

then on the next slide he showed how to get the matrix [[5 , 0], [2, 4]] when we know the individual matrices. We reverse the order in which we apply these matrices:

Now going by the same analogy when we formulate the concept of the matrix inverse, we know how we go from basis vectors to the 1st transformation and then we want to apply the 2nd transformation so that we get back the basis vectors. The slide shows the setup:

Note that in this case we know the result of the composition and we want to identify the matrix for the 2nd transformation. So far the setup is same as

What i do not understand is that when looking for the inverse why we do not reverse the order in which we multiply matrices and instead write it as shown on this slide:

For the matrix on this slide , i will get the same answer if i reverse the order as the 1st matrix is symmetric but my question is what is the general approach. Should we not write this as :

[[a ,b], [c, d] ] * [[3,1],[1,2]] = [[1,0][0,1]] and then solve the system of linear equations even when the matrix for which we want the inverse is not symmetric. Going by the instructor’s explanation it is the application of the transformation given by the inverse matrix on the original matrix gives us the effect that we are back to the basis vectors.

Please comment.

Thanks a lot in advance