Yeah! :D
This is how I think with matrices very smoothly and easily. Try to think in terms of the actual picture. The matrices MUST follow because that's what they represent as long as you have an accurate image in your mind of what these things like "rotation" or "shear" actually mean.
For instance, if you never saw a matrix before but were introduced to the idea of commutativity, you could fairly easily know without proof that for a matrix to represent rotation in the 2D plane, it would have to commute with other 2D rotation matrices, even though in general matrices don't commute.
Why does this not need proof I claim? Because it would be absurd since it's essentially putting the cart before the horse. The rotation is what we want, and the act of rotation commutes, this is what we demand of our matrix. If matrices didn't satisfy this behavior, we would have thrown them away and found something better that does commute like we need! The numbers are simply details that follow for anyone who wishes to calculate exact values of a certain case for engineering perhaps.
I think more fundamentally we should think of numbers (real, complex, quaternions) as really special cases of matrix and vector multiplications since all numbers can be really seen in a more general view as representing both matrices and vectors simultaneously.