Proof of Theorem DERC

Suppose that $A$ is a square matrix of size $n$ where the two rows $s$ and $t$ are equal. Form the matrix $B$ by swapping rows $s$ and $t$. Notice that as a consequence of our hypothesis, $A=B$. Then \begin{align*} \detname{A} &= \frac{1}{2}\left(\detname{A}+\detname{A}\right)\\ &= \frac{1}{2}\left(\detname{A}-\detname{B}\right) &&\knowl{./knowls/theorem.DRCS.knowl}{\text{Theorem DRCS}}\\ &= \frac{1}{2}\left(\detname{A}-\detname{A}\right) &&\text{Hypothesis, $A=B$}\\ &= \frac{1}{2}\,(0)=0 \end{align*}

The proof for the case of two equal columns is entirely similar, or could be derived from an application of Theorem DT employing the transpose of the matrix.