which implies and, if you consider the fact that exchanging two rows of a matrix in succession will bring back the original matrix, then this result is obvious. Coming back to the construction of the transformation matrix for the linear system of equations under consideration, matrix is now constructed as follows

This approach of constructing the transformation matrix demonstrated using a matrix can be easily generalized for a system of equations involving an matrix . For example, elementary matrix which reduces element in matrix to zero, can be constructed by simply inserting at location in an identity matrix, i.e. --------(21)while the permutation matrix which interchanges i'th and j'th rows, can be created by interchanging i'th and j'th rows of the identity matrix. The transformation matrix for a general matrix can then be constructed by multiplying the elementary matrices, and the permutation matrices, in appropriate order such that

It is important to note that the explanation presented in this subsection provides insights into the internal working of the Gaussian elimination process. While performing the Gaussian elimination on a particular matrix through a computer program, neither matrices ( nor matrix is constructed explicitly. For example, reducing elements in the first column of matrix to zero is achieved by performing the following set of computations where Performing these elimination calculations, which are carried out row wise and may require row exchanges, is equivalent to constructing matrices ( and effectively matrix which is an invertible matrix. Once we have reduced to form, such that u for all i, then it is easy to recover the solution using the back substitution method. Invertibility of matrix guarantees that the solution of the transformed problem is identical to that of the original problem.