Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 1 / 6D / a / Solution

Use columnwise partial pivoting: select the largest absolute entry in the current column and interchange rows. In the first column the pivot is $4$, so interchange rows 1 and 2. The multipliers are $1/2$ and $-1/2$, and elimination givesThe second pivot is $5/2$, so interchange the two remaining rows. Their previously stored first-column multipliers must also be interchanged. The final multiplier is $(1/2)/(5/2)=1/5$, giving the LU decompositionThe permutation matrix orders the original rows as $2,3,1$. Multiplication verifies the displayed identity.

If “column pivoting” instead denotes interchanging columns rather than searching within a column, that convention gives $AQ=\tilde{L}\tilde{U}$ withThis second identity uses a different pivot convention; it does not change the row-pivoted answer above.