= Solution
Let $e_i$ be standard coordinate columns. The required <elementary matrix> is
$$
\boxed{E=I_n+\alpha e_je_i^T,\qquad EA:\ R_j\gets R_j+\alpha R_i.}
$$
It 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 $\det(E)=1$ and
$$
\boxed{\det(EA)=\det(E)\det(A)=\det(A).}
$$
Hence adding a multiple of one row to another leaves the <determinant> unchanged, even if $A$ is singular.
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