# About the definition of a singular and non-invertible matrix

In Wolfram Mathworld, they defined a singular matrix as:

A square matrix that does not have a matrix inverse.

So my questions regard this definition are:

1. Why is the definition of singular matrices restricted to square matrices? Aren't all rectangular matrices singular and non-invertible since they don't have inverses?
2. When talking about matrix inverse, does it only refer to two-sided inverses and not one-sided inverses?
3. Can the terms singular and non-invertible be used interchangeably?
4. The invertible matrix is defined as having a determinant not equal to zero. Do rectangular matrices also have determinants and can this rule be applied to rectangular matrices as well?