Why is Gaussian elimination still being taught when there are more efficient methods?
By efficient methods I mean less time and effort consuming ways to solve a system of linear equations. I have nothing against learning. I´m just really curious about this.
The problem is that the more efficient methods are more complicated and are more efficient only for large matrices.
The original faster method was Strassen's.
Here is a typical paper. found by a Google search for "solving linear system of equations strassen", discussing the effectiveness of using Strassen's method:
http://www.ibspan.waw.pl/~paprzyck/mp/cvr/research/varia_papers/CMA_S_96.pdf
It says that Strassen's method was faster for $n=150$. For $n=1500$, Strassen's method was $30%$ faster.
This paper was written in 1996, so more recent results are almost certainly available.