Performing the LU decomposition

The Gaussian elimination also enables us to decompose the matrix in (1.46). We already have the upper triangular in (1.49). To form the permutation matrix P we will interchange those rows of the identity matrix I that have been interchanged in A in the course of the Gaussian elimination. Let (i,k ) denote the operation of interchanging rows i and k in the i-th step, then what we did is (1,1), (2,4) and (3,4). These operations applied to the identity matrix I result in the permutation matrix 

The lower triangular matrix L can be constructed from the multipliers used in the elimination steps if we adjust them according to the rows interchanged. Taking into account that for the row of the pivot the multiplier is necessarily 1.0 (i.e., this row remains unchanged), in the three steps of the Gaussian 

In step 3 the interchange was (3,4), which will affect all the previous multipliers, resulting in (1.0, 0.0, 0.4, -0.6) and (1.0, -0.2, 0.8), whereas (1.0,-0.5), used in this last step, remains unchanged. We put these vectors into the lower triangular of a matrix and obtain 

Now we present a module for the LU decomposition and apply it to compute the determinant of A. As is well known, det(A) is a number, defined by 

Since det(A) =0 if and only if A is singular, it provides a convenient way of checking singularity. Determinants have traditionally been used also for solving matrix equations (ref. 10), but both the Gauss-Jordan method and the Gaussian elimination are much more efficient. The determinant itself can easily be calculated by LU decomposition. For the decomposed matrix (1.45) 

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