My question is why we have to restrict over selves to find inverse for non singular matrix?
We can define inverse of a square matrix as follows:
A matrix $B$ is said to be inverse of $A$ if $BA = C$, where $C$ is the matrix obtained by $A$ by applying row transformation (some what like normal form). Matrix $C$ must satisfy following properties:
- All zero rows are at the bottom
- leading entry of each non-zero row is $1$.
- $C$ is the matrix obtained by applying row transformation to the maximum extent. The matrix $C$ is identity matrix if $A$ is non singular. Only in that case, $BA = AB = I$ holds.
In this way we can associate with every square matrix a matrix $B$ (inverse) and $C$ (some what normal form of $A$). This will help to solve system of equations even when they have infinite solutions! Am I right?