I recommend figuring out how to make a video about scattering matrices that is exciting. Let me know and I will do it!

Somehow I missed this comment. Very sorry! I suspect the following... It is very rare in numerical methods to calculate the inverse of a matrix. Something like A^-1 B is not the inverse of A times B. Instead, look at this as matrix division. In this case, it is a predivision. So the code in MATLAB would be A\B instead of inv(A)*B. The matrix division does not calculate the inverse and will run orders of magnitude faster and more efficiently than if a matrix inverse is calculated. Is this what you are seeing?

In the general case, it is a fields problem. The fields on one side of an interface must match the fields on the other side. A change in impedance being responsible for reflections assumes the transmission line is otherwise the same. However, if you hook a 50 ohm micro coaxial line up against a 50 ohm parallel plate line, you will get reflections because the electromagnetic fields look different in the lines.

It appears we goofed. Thank you for catching the mistake!! We will fix this ASAP.

Me too!