Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Digital systems, linear algebra, introduction to formal proof / 5 / a

Left multiplication by an elementary matrix adds a multiple of row one to its designated target row. The first-column entries below the pivot are $1,2,-1$, so $a=-1$, $b=-2$, $c=1$. The corresponding row operations are $R_2\leftarrow R_2-R_1$, $R_3\leftarrow R_3-2R_1$, and $R_4\leftarrow R_4+R_1$. Since none changes row one, their sequential application gives the required three eliminated first-column entries and the displayed remaining entries.

The second pivot is two. Row three has second entry two, so subtract row two; row four has second entry minus two, so add row two. Therefore $d=-1$, $e=1$. Their trailing row pairs become $(0,4)$ and $(3,5)$, respectively; the zero in the third pivot position is why a later row interchange is needed.

Let $e_{ij}$ denote the matrix unit with one at position $(i,j)$ and zero elsewhere. The five inverses areThus each is the identity matrix with only the indicated off-diagonal matrix entry altered. These inverses reverse the corresponding row additions; no row scaling or permutation is involved.