Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Digital systems, linear algebra, introduction to formal proof / 5 / c

First subtract row one from row two, using $E_1$; then subtract four times the updated row two from row three, using $E_2$:The updated second row is $(0,1,2)$ and the final third row is $(0,0,-4)$. Hence $E_2{E}_{1}A=U$ and $L=E_1^{-1}{E}_2^{-1}$. The resulting LU decomposition isReversing the order when undoing the elementary operations is essential.

Introduce $z=Ux$. Solve $Lz=b$ by forward substitution and then $Ux=z$ by back substitution. For these factors, the recurrences arefollowed byThe nonzero diagonal entries make both triangular solves well-defined. No matrix inverse is calculated.