We have shown in STEP 11 how to solve a linear system Ax = b by Gauss elimination, applied to the augmented matrix . In the preceding Step, we have extended the elimination process to calculate the inverse A^<-1 of the coefficient matrix A, assuming it exists.

Another general approach to solving Ax = b is known as the method of LU decomposition, which provides new insights into matrix algebra and has many theoretical and practical uses. Efficient computer algorithms for handling practical problems can be developed from it.

The symbols L and U denote lower triangular matrix and upper triangular matrices, respectively. Examples of lower triangular matrices are

Note that in such a matrix all elements above the leading diagonal are zero. Examples of upper triangular matrices are:

where now all elements below the leading diagonal are zero. The product of L₁ and U₁ is