Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as multiplication by permutation matrices.
However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations.
Other times these additional operations are worthwhile because they add numerical stability to the final result.
Furthermore, in Gaussian elimination it is generally desirable to choose a pivot element with large absolute value.
The following system is dramatically affected by round-off error when Gaussian elimination and backwards substitution are performed.
In this case it is desirable that we interchange the two rows so that a21 is in the pivot position Considering this system, the elimination algorithm and backwards substitution using four-digit arithmetic yield the correct values x1 = 10.00 and x2 = 1.000.
Complete pivoting is usually not necessary to ensure numerical stability and, due to the additional cost of searching for the maximal element, the improvement in numerical stability that it provides is typically outweighed by its reduced efficiency for all but the smallest matrices.
This strategy is desirable when entries' large differences in magnitude lead to the propagation of round-off error.