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

The required elementary matrix isLeft multiplication exchanges the two rows. Applying the same exchange twice restores every row, so $E^2=I_3$.

The elementary row operation and its reverse areThus $FA$ replaces row three by row three minus three times row two. The inverse adds those three copies back.

Use multiplicativity of the determinant and the determinant criterion for an invertible matrix:Neither factor can therefore be zero, so both matrices are nonsingular. The determinant of a triangular matrix is the product of its diagonal entries:Each product is nonzero, forcing every diagonal factor to be nonzero. These explicitly stated determinant properties justify both conclusions.

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.

With column-vector homogeneous coordinates, the translation matrix isIt maps $(x,y,1{)}^T$ to $(x+a,y+b,1{)}^T$. The final coordinate incorporates the constant displacement into matrix multiplication.

Reflection in $x=0$ sends $(x,y)$ to $(-x,y)$. Therefore its homogeneous-coordinate matrix isIts square is the identity, as required for applying the same reflection twice.

Translate the mirror line to zero, reflect there, then translate back. Under the column-vector convention the rightmost operation acts first:This composition of affine transformations maps $(x,y)$ to $(4-x,y)$, fixing exactly the points with $x=2$.