M^-1*v and v*M^-1 is not equal too
It can't be!!
You cannot multiply a n*1 vector with n*n matrix!
But you can multiply a n*n with a n*1 or a 1*n with a n*n as shown.
And yes, I agree about the distinction between 0 and 0V. It would be convenient not having to specify units in that case.