Gaussian Elimination is of course essential to Linear Algebra, as I note having taken three introductory mathematics subjects (at Higher level) at university so far.
But too often, at least at my university, it becomes central to one's learning of Linear Algebra even in second year, often at the expense of truly understanding what is going on. (For instance, how matrices transform vector spaces.)
Being able to row reduce under exam pressure then becomes essential to performing well in the subject, even though at higher levels, the computations are merely a means to an end, that is, an interpretation of the result of a reduced system. (For instance, to determine the kernel of a subspace, or whether a line and plane intersect.)
I would like to know the opinion of those who have completed mathematics study at higher levels, whether or not Gaussian Elimination remains an essential part of later assessments (particularly in exams), and if so, whether a better alternative for assessment would be simply to provide the original system, as well as the row-echelon and reduced row-echelon forms for that system, so that it may be interpreted without wasting time on performing computations that are often outsourced to computers outside an exam room.
I feel that it would be better to examine row reduction only in first year, when it is first learnt, so that students know how to perform it themselves, and in later years, having the results of a computation already performed for them.
The real task is in understanding what the system represents, which is a higher order skill. So for instance, a student could be asked a question about determining whether a company under certain conditions can meet its production targets, and then being asked to write equations representing this, and then using a provided system in RREF on another page to interpret the results and hence answer the questions.
A harder question could be one involving whether or not a certain output is possible, which would rely on the student's knowledge of subspaces like the image of a matrix transformation, or in finding a least squares solution to a system.
Please let me know your thoughts on this idea. (If there is a more suitable Stack Exchange for this forum, let me know and I'll remove the question from here.)