Theorem 3. 10(Direct Factorization A=LU. No Row Interchanges Suppose that Gaussian elimination, without row interchanges, can be suc cessfully performed to solve the general linear system AX= B. Then the matrix A can be factored as the product of a lower-triangular matrix L and an upper-triangular matrix U A= LU Furthermore l can be constructed to have 1's on its diagonal and u will have nonzero diagonal elements. After finding and u the solution x is computed in two steps 1. Solve lu =b for y using forward substitution 2. Solve ux=y for X using back substitution