# Can a matrix be invertible if it is not square?

##### 1 Answer

#### Answer:

No

#### Explanation:

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.