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 / 5 / b / Solution

Let $e_i$ be standard coordinate columns. The required elementary matrix isIt is the identity matrix with the extra entry $\alpha$ in position $(j,i)$. Left matrix multiplication implements the row operation: all other rows are unchanged. Since $j>i$, $E$ is a lower triangular matrix with every diagonal entry one. Part (a) therefore gives $\det (E)=1$ andHence adding a multiple of one row to another leaves the determinant unchanged, even if $A$ is singular.