Comment by jules

Yeah, I agree that most questions in introductory linear algebra are reducible to Gaussian elimination. However, if I had to play devil's advocate, I'd say:

1) Gaussian elimination is only one of the algorithms for solving linear equations. Why that one? Perhaps there is a better one for pedagogical reasons.

2) Why focus on the algorithm in paricular? Maybe we could formulate the thing Gaussian elimination is doing more abstractly with less reference to a particular algorithm. For instance: any matrix A can be written as LDU where L is strictly lower triangular, D is diagonal, U is strictly upper triangular. Or maybe: any linear map can be written as a projection on the first k coordinates relative to some basis, where k is the rank. Or maybe there is a geometric way to understand it.

3) Why focus on this particular concept in particular? That concept is just one of many concepts in introductory linear algebra. Although you can reduce everything in introductory linear algebra to it, you could also pick some other concept around which you could center the course, such as linear independence, the determinant, or something else. Or some other decomposition, such as A = XDY where X,Y are products of elementary row operations (so they are square and have determinant 1), and D is a rectangular diagonal matrix. This decomposition is arguably more important than the LDU decomposition associated to Gaussian elimination. Why even focus on one concept in particular?

4) This is only for the very basics. You also need a plan for Gram-Schmidt, eigenvalues, spectral theorem, Jordan decomposition, and so on, which form the meat the linear algebra courses.