[[Answered before "invertibility" was added to $A$.]] I think some assumptions are missing here. It's either impossible or trivial.

OOH, if you're forbidding $i=j$ in your matrices, you can let $A=\pmatrix{1&1\\ 0&0}$. No matter which row operations you use, you will always get a nonzero matrix where the two columns are the same.

OTOH, if you're allowing elementary matrices where $i=j$, then it's trivial: $E_{1,1}(0) \cdot E_{2,2}(0) \cdot \cdots \cdot E_{n,n}(0) = 0$, and $0A=0$ definitely is diagonal.