This video generalises the LU-decomposition from before to non-square matrices where also row exchanges could be needed. This means that the PLU decomposition is the general procedure to apply the Gaussian elimination in a closed matrix form.
PLU decomposition - An Example
The PLU decomposition for non-square matrices will be explained with the help of an example. Essentially the whole procedure is the well-known Gaussian elimination but here we also save all the steps in two matrices. The permutation matrix $P$ is only used to store row exchanges.