Another convenient method of solving systems of equations is by decomposition of the matrix into a product of upper and lower triangular matrices and solution of the system by combined forward and backwards substitutions. One has the product of two matrices, each with six elements. However, the original matrix has only 9 elements, whence one can set the diagonal elements of one of the triangular matrices equal to 1, for example, the diagonal elements of the upper matrix:
Note now that there are 9 equations expressing the aij in terms of the lij and uij. A special feature of this system is that one can find the solution explicitly and that, once an aij has been used, it can be overwritten. For this purpose, one need only follow the proper sequence:
The original system can now be rewritten:
and one can set
Hence one has for the yi the system:
which can be solved by forward substitution. Once the yi have been found, the xi are determined by backward substitution from the system defining the yi .
whence the system for the auxiliary variables yi is:
i.e., y1 = 3.75, y2 = -(8 +3*3.75)/2.5 = -7.7, y3 = (3 - 3.75 - 7.7/2)/1.8 = - 2.5555, and the xi are the solutions of the system
i.e., x1 = -5.5555, x2 = -7.7 .7, x3 = (3-3.75+7.7/2)/1.8 = 1.72222.
