Given an $n \times n$ matrix, and row operations shear, switch, and scale, what is the largest number of efficient row operations required to row reduce to echelon form?
For reference, a shear is defined as $R_i \rightarrow R_i + \lambda R_j$, a switch $R_i \leftarrow \rightarrow R_j$ and a scale, $R_i \rightarrow \lambda R_i$ for non-zero $\lambda$.
I've claimed that $n^2$ is an upper-bound for efficient operations, but never proven. So, I'm curious, how would one determine the best set of moves? I would ask about the least amount, but that is highly dependent on the composition of the matrix. Specifically, if the matrix is already in reduced-row-echelon form then no moves are required. Even supposing that the matrix isn't, the ideal scenario is we have to perform 1 operation. Hence, a greater motivation for the maximum of efficient moves instead of least amount of moves.
A possible application of this result could aid students in understanding if row-reduction of an arbitrary $n \times n$ matrix is incorrect based on length. For example, if $n^2$ is held true, a $2 \times 2$ matrix should take no more than 4 efficient operations to solve.
If the matrix elements are algebraically independent, each row operation can only produce one more $0$ or $1$. For an $n \times n$ nonsingular matrix, you want $n(n-1)/2$ $0$'s and $n$ $1$'s, so this will take at least $n(n+1)/2$ row operations.