us to treat the same way as we did with $A^{(0)/math>,}} repeatedly. If dimension of is , after such steps all columns form sub-diagonal part of triangle matrix and all pivots combined with rows form upper triangle matrix U, as required. In the above example so only two steps suffice.$

The above procedure demonstrates that at no step the top diagonal pivot element of consecutive sub-matrices can be zero. To avoid it columns or rows may be swapped so that becomes nonzero.Such procedure involving permutation