How to find the inverse of a matrix?

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More specifically I mean, when we use row operations to find the inverse of matrix, we start by writing $(I\,|\, A)$, where $A$ is a matrix and $I$ is the identity matrix.

But when we use column operations to find the inverse of matrix then we start by writing $(A\,|\,I)$.

Why is there this difference and what does it mean? Please explain.