Question

Can a matrix be invertible if it is not square?

Answer 1

No

Explanation:

Let's consider a non-square matrix that is dimension `M` x `N` where `M ne N`. This means that the matrix maps an N dimensional space to an M dimensional space.

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 `M < N` (if `M>N`, we can talk about the inverse instead, so this isn't a bad thing to do). That means that we're taking a higher dimensional space and mapping it to a smaller dimensional space.

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.