Go beyond 2 x 2 and row echelon form is the way to solve systems of linear equations.
This stuff is not done by hand, but by a computer. Row echelon form is much, much faster than
using determinants (Cramer's formula). Ask someone who works in numerical analysis about this. Your friend with the Ph.D. is clearly unaware of the needs of applied math.
In pure math, Cramer's formula is very important for some proofs (and I personally prefer it for conceptual reasons over the algorithmic row echelon business, but then I don't work in numerical analysis). In applied math problems that give rise to systems of linear equations, you typically face equations in hundreds of variables. You are not going to solve that with a determinant. Moreover, if a computer needed to find a determinant of a 100 x 100 matrix it would not use the sum of 100! terms as in the Laplace expansion formula for determinants, but instead apply faster algorithms (the QR algorithm).
Your profile says you are studying economics. At some point when you may need to deal with some system of linear equations in economics and have the computer solve it for you, the method of solution could be treated as a black box in the same way most people drive without knowing how a car really works. But the reality is that if it were not for the algorithm of reducing to row echelon form, the computer couldn't solve systems of hundreds of equations rapidly.
In addition to lessening the workload involved in computing a determinant of a square matrix, e.g.,
- one can confirm or rule out whether a square matrix is invertible by stopping before full reduced row-echelon form,
- we can determine the rank of a matrix without needing to go through the tedious work sometimes involved in obtaining full reduced form row-echelon form, and
- we can determine whether the row or column vectors of a matrix are linearly dependent or independent with just plain old row-echelon form.
Certainly, each form has its uses, and the fact that we can sometimes avoid extensive and tedious computations (which also leave room for introducing simple errors along the way), that's not to say that it is not important to know how and when to obtain reduced row echelon form: but we can "pick-and-choose" to some extent "how far" we need to row-reduce.
Best Answer
For the resolution of linear systems, Gaussian elimination is preferred over Gauss-Jordan (these parallel the echelon and reduced echelon forms) as the former involves less operations (roughly $n^3/3$ vs $n^3/2$), for a similar numerical stability.
Anyway, Gauss-Jordan does not require a backsubstitution step, which makes the code a little more compact. On the other hand, the reduced form is not compatible with LU decomposition.
From a theoretical point of view, the reduced form has the advantage of being uniquely defined for a given matrix.
Anyway, it is worth to note that for an invertible matrix, the reduced form is simply a unit matrix, totally uniniteresting. The concept is only useful for degenerate cases.