Let $A$, $B$, $X$ be matrices of the same order such that $X = AB$.
When applying an elementary row operation to the matrix equation of $X = AB$, then the elementary row operation is applied to $X$ on the left side and only to $A$ on the right side. Why is the elementary row operation in above scenario not applied to the matrix given by the product $AB$ but only to the first matrix $A$ when applying it to left side?
My understanding is that if two matrices $X$ and $Y$ are equal i.e. $X = Y$, then in order to maintain equality we must apply the same elementary row operation to $X$ and $Y$. In the question being posted $Y$ is like $AB$, and it follows that the row operation must apply to the product $AB$ rather than just $A$.
An example is as given below.
