Can a matrix be invertible if it is not square?
Let's consider a non-square matrix that is dimension
Logically, its inverse therefore maps an M dimensional space to an N dimensional space.
Since the inverse of the inverse is the original, let's just assume
By definition of a dimension (and linear span and eigenvalues), this cannot be invertible.
One can construct a trivial operator that can be inverted (such as x -> (x, 0), i.e. the matrix [1, 0]), but those represent a lower dimensional space, such as a plane or a line within 3-space.