Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 1 / a (Row reduction)

The three elementary row operations on an $m\times n$ real matrix are:

Each operation changes whole rows, so it acts on every column by the same rule. Each can be undone by an elementary row operation.

For an $m\times n$ matrix, the corresponding elementary matrices are $m\times m$, because the operation is implemented by left matrix multiplication. Apply the desired elementary row operation to the identity matrix $I_m$. Writing $E_{ij}$ for the matrix unit with its only nonzero entry, $1$, in position $(i,j)$ gives:Indeed, these matrix products interchange rows $i,j$, scale row $i$, and add $c$ times row $j$ to row $i$, respectively. The displayed inverses undo precisely those operations. Algebraically, $E_{ij}^2=0$ when $i\ne j$, so $(I_m+c{E}_{ij})(I_m-c{E}_{ij})=I_m$; the other two inverse identities follow immediately from their row actions. Thus every elementary matrix is an invertible matrix.

Perform the elementary row operations $R_2\leftarrow R_2+2R_1$ and $R_3\leftarrow R_3-R_1$, then $R_2\leftarrow R_2/2$, $R_1\leftarrow R_1-R_2$, and finally $R_1\leftarrow R_1+R_3$, $R_2\leftarrow R_2-3R_3$:Consequently the reduced row echelon form is $I_3$. Applying the same operations to the augmented column $b=(b_1,b_2,b_3{)}^T$ givesThus every $b\in {\mathbb{R}}^3$ admits exactly one solution. This follows because the elementary row operations preserve the solution set of the augmented linear system, and the reduced coefficient matrix is the identity matrix.