Proof of Theorem DZRC

Suppose that $A$ is a square matrix of size $n$ and row $i$ has every entry equal to zero. We compute $\detname{A}$ via expansion about row $i$. \begin{align*} \detname{A} &= \sum_{j=1}^{n}(-1)^{i+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}} &&\knowl{./knowls/theorem.DER.knowl}{\text{Theorem DER}}\\ &= \sum_{j=1}^{n}(-1)^{i+j}\,0\,\detname{\submatrix{A}{i}{j}} &&\text{Row $i$ is zeros}\\ &= \sum_{j=1}^{n}0=0 \end{align*}

The proof for the case of a zero column is entirely similar, or could be derived from an application of Theorem DT employing the transpose of the matrix.