The inverse of a matrix is another matrix such that when they multiply together the identity matrix is the result. A matrix might not have an inverse, and if that is the case that matrix is called singular. If some matrix A does have an inverse that is typically denoted using a superscript of -1 ie. A^-1 is the inverse of A. We can then write 

A^-1 A = I 

where I is the identity matrix of some suitable size. What would be that size ?

Suppose that matrix A has m rows and n columns. We usually say A is an m by n matrix, A is an m times n matrix or even briefer A is m x n. For any matrix multiplication to work the number of columns of the matrix on the left side must equal the number of rows of the matrix on the right side. Hence the inverse of A must have n rows and m columns. We would write that A^-1 is n x m.

A^-1 A = I

(n x m) (m x n) = n x n

A A^-1 =I

(m x n)(n x m) = m x m

So where A is non-square ( m <> n )