We have already seen that factoring matrices into simpler ones is important in developing and applying matrix theory. In this chapter we look at factoring an m × n matrix either as LU (where L is a lower triangular matrix and U a row echelon form) or as QR (where Q is an orthogonal matrix and R a row echelon form). Both factorizations involve a kind of Gaussian elimination approach and are highly used in numerical algorithms and software, such as MATLAB. Knowing this material also helps us understand the occasional warnings given with a MATLAB computation.