The most common method to find the inverse of a invertible square matrix is to apply row operations to the matrix and reduce it to the identity matrix.
An row operation is applied to the matrix by doing a left multiplication by an elementary matrix.
Denote the elementary matrices as $E_1, E_2, ... , E_n$ , then
$(E_1, E_2, ... , E_n)A = I$ , where $E_1, E_2, ... , E_n$ is the inverse of $A$.
However, the procedure to check whether two matrices $A$ , $B$ are the inverse of each other is to check that $AB=I$ and $BA=I$.
Why do we claim that $(E_1, E_2, ... , E_n)$ is the inverse of $A$ without checking that $A(E_1, E_2, ... , E_n) = I$ ?
