row echelon vs reduced row echelon form

Question

I apologize if this is a very basic question.

I understand the difference between the two forms, but i was curious when row echelon from is enough. where is row echelon form used?. Why shouldn't I always go for reduced row echelon form?

Answer 1

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.

Answer 2

One example: if using row operations to simplify the calculation of a determinant. It is just as easy to compute the determinant of an upper triangular matrix as to compute the determinant of a diagonal matrix; so, there is no reason to go all the way through to RREF.