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Elementary row operations on a matrix

I'm new to Mathematica and am teaching an applied linear algebra course.  We start off doing elementary row operations on an augmented matrix to solve a system of equations.  I know about the RowReduce command, but that does all the row operations at one time.  I'd like for my students to be able to do them one-by-one so that they get the experience of thinking about what they should do next, but don't get bogged down in the arithmetic.  So, my question is: what is the easiest way to do each of the three types of elementary row operations?  (The Maple version is described here:
POSTED BY: Betty Love
The function Part can be useful, because it allows access to the rows and columns of a matrix. For example, mat[[1]] returns the first row, and mat[[All, 1]] is the first column. Assignments are also possible, e.g. mat[[1]] = mat[[1]] - 2 mat[[2]] will modify the matrix in place by subtracting twice the second row from the first one.

This tutorial from the documentation may be of interest, as well as this related discussion on the Mathematica StackExchange site.
POSTED BY: Ilian Gachevski
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