First, students will review what identity matrices are. After that, students will learn what an elementary matrix is. That is, an elementary matrix \(E\) is an \(n \times n\) matrix that can be obtained from the identity matrix \(I_n\) by one elementary row operation. Then students will learn that all elementary matrices are invertible, and that the inverse of an elementary matrix is also an elementary matrix. After that, students will learn how to determine whether a given matrix is elementary. Here's a lesson, and a good source of practice problems.
TODO: Prove the two properties mentioned.