I don’t think it is possible for me to cover a whole section of this linear algebra book in one class. Today, we started the section on Elementary Matrices and just a little ways in, I knew it was going to be a difficult section.
The main purpose of this section is to use the concept of matrix multiplication to perform row operations and demonstrate the basic results of row equivalence, such as categorizations of nonsingularity: (i) that nonsingular matrices are row equivalent to I, (ii) that the system has a unique solution. Also, elementary matrices provide a way to calculate matrix inverses through row reduction.
However, a good many of the results require a level of proof that is new to the students. I know that many of them are planning on going into engineering, another group is planning on teaching either middle school or secondary mathematics, and last of all, a group will likely go to graduate school in mathematics. It’s tough to design a course in linear algebra to meet all their needs but in the end, it’s worthwhile to see the reason behind the method.