Define the Gaussian complexity of an $n \times n$ matrix to be the minimal number of elementary row and column operations required to bring the matrix into upper-triangular form. This is a quantity between $0$ and $n^2$ (via Gaussian elemination). The notion makes sense over any field.
This problem certainly seems very basic and it must have been studied. Surprisingly, I don't know of any references. So, I'll be happy with any reference there is. But, of course, the main question is:
Are there any non-trivial explicit lower bounds known?
By nontrivial I mean superlinear. Just to be clear: Over finite fields a counting argument shows that a random matrix has complexity order n^2 (a similar claim should be true over infinite fields). Hence, what we're looking for is an explicit family of matrices, e.g., Hadmard matrices. This is the same as with Boolean circuit complexity where we know that a random function has high complexity, but we're looking for explicit functions with this property.