I actually think Julia's behaviour (partially) copied from matlab is bad. I had to review a lot of matlab/Julia simulation code write by PhD/grad students (as well as my own) and most of the time when things somehow are not working debugging becomes a task of "find the missing dot". If they would have decided on a more obvious symbol (the dot is really just way too easy to overlook) for broadcasting it would be much better, but I still much prefer that broadcast and matrix operation would use entirely different operators/named functions. Instead we have the case that a little annotation can dramatically change the behaviour of the code (the matrix exponential is in fact a great example).

That said I mainly work with fields and calculus so I might be biased because the majority of operations are broadcast and I only rarely use matrix operations.