Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof (resit) / 5

An elementary matrix acts on rows by left multiplication. In the stated order, the three matrices areThe first scales row 3, the second adds $-3$ times row 2 to row 4, and the third swaps rows 1 and 4. Right multiplication would instead perform column operations.

Undo each elementary row operation: divide row 3 by 3, add 3 times row 2 to row 4, and repeat the same swap. HenceMultiplying each inverse by its original matrix gives the identity matrix.

First add row 1 to rows 2 and 3. This is the product of two elementary matrices, written compactly asSwap rows 2 and 4 with $P$, then subtract half of the new row 2 from row 3 with $E_2$:This gives the upper triangular matrixBecause $P^{-1}=P$, rearranging yields the convention $A=PLU$, whereThe conjugation by $P$ moves the earlier elimination multipliers to the appropriate rows. Ignoring this permutation when forming $L$ would give an incorrect PLU factorization. Here the same permutation is its own inverse, so $PA=LU$ also holds.

The determinant of the unit lower triangular matrix $L$ is 1. There is one row swap, so $\det P=-1$, and the upper triangular matrix has determinant $1\cdot (-2)\cdot (3/2)\cdot 2=-6$. Therefore