Here is the examples page on Wolfram|Alpha that lists some matrices operations. Hope that helps!

-Adriana

For the purpose of getting a consistency requirement, one can augment the matrix on the right with an identity matrix. Here is your example.
In effect the last columns give the requirement that, for the indicated pivots to be correct, 2a-b not be equal to zero.

Explaining this to students first encountering the subject might be another matter though.

Another possibility, which might or might not work reliably, is to use LUDecomposition.

probably you noticed, but W|A will show you the row operations "step-by-step" if you ask to row reduce a matrix

Thanks for the ideas. I have my linear algebra students use C ormullion's approach in Mathematica. But not all of them have laptops to bring to class, so I was hoping to do something similar in Wolfram Alpha during class time. Sam's approach is a bit too advanced for my students who may have no programming experience, particularly Mathematica programming. I suppose I could teach that, but would rather not add complexity.
I would be ok with just using the RowReduce command, which works in Alpha, except that it reduces too far (treats parameters as non-zero constants) for some of my problems.  For example, I sometimes ask my students to find all values of certain parameters for which a system of equations is consistent. RowReduce doesn't work for this as this snippet shows:gives meWhat I want isso that students can see that the system is consistent when -2a+b=0.

There might be quicker ways, but I know you can operate on rows and columns of lists:
I tend to create new tables rather than modify the existing ones, as here. It's probably safer for modestly-sized tables...