Question A.11: Consider the matrix A = |1 2 1 0 0 3 3 1 2 0 2 0 1 0 0 2| It......

Add an identity matrix, which will ultimately be converted into a U matrix and the \overline{A} matrix will be converted into an L matrix:

First step, p = 1:
Table 1 and Table 2

The shaded columns and rows are converted into L and U matrix column and row and the elements of \overline{A} matrix are modified using Equation A.101, that is,
l_{ip} = a_{1}p  for ≥ p
u_{pj} = \frac{a_{pj}}{a_{pp}}  for j > p                                 (A.101)
a_{ij} = a_{1j} – l_{ip}u_{pj}  for i > p,  j > p

a_{32} = a_{32} – l_{31}u_{12}
= 0 – (2)(2) = -4
a_{33} = a_{33} – l_{31}u_{31}
= 2 – (2)(1) = 0

Second step, pivot column 2, p = 2:
Table 3 and Table 4

Third step, pivot column 3, p = 3:
Table 5 and Table 6

This is the same result as derived before in Example A.10.