swap the first row with the row having the maximal absolute value element in the first column make every other element in the first column equal to by adding appropriate multiples of the first row to the other rows suppose that at has plu decomposition pt lt ut then it is easy to check that ## at > pt- ( pt > ct /at lt ut { { { pt- lt- ut- is plu decomposition of at- since the plu decomposition for the scalar an- is trivialby working backwards we obtain plu decomposition of example (plu decompositiontake - our goal is to modify via steps and above so as to obtain an upper triangular matrix with maximal elements on the diagonal we first swap the first and second row nextwe add - / times the first row to the third row and / times the second row to the third row - - - - - - - / / the final matrix is and in the process we have applied the permutation matrices using the recursion ( we can now recover and namelyat the final iteration we have and / and subsequently - / / - / observing that [ ] - and / plu decompositions can be used to solve large systems of linear equations of the form ax efficientlyespecially when such an equation has to be solved for many different this is done by first decomposing into pluand then solving two triangular systems ly pb ux