Answer
Let $S(X)=XE$ be a matrix transformation
If $S$ is equivalent to one row replacement, then
\begin{align*}
& \det E=\det I = 1
\\ & \det{AE}=\det{(S(A))}=\det A
\\ & \Rightarrow \det{AE}=(\det E)(\det A)
\end{align*}
If $S$ is equivalent to one row interchange, then
\begin{align*}
& \det E=-\det I = -1
\\ & \det{AE}=\det{(S(A))}=-\det A
\\ & \Rightarrow \det{AE} = (\det E)(\det A)
\end{align*}
If $S$ is equivalent to one row scaling of $k$, then
\begin{align*}
& \det E=k\det I = k
\\ & \det{AE}=\det{(S(A))}=k\det A
\\ & \Rightarrow \det{AE} = (\det E)(\det A)
\end{align*}
Work Step by Step
Let $S(X)=XE$ be a matrix transformation
If $S$ is equivalent to one row replacement, then
\begin{align*}
& \det E=\det I = 1
\\ & \det{AE}=\det{(S(A))}=\det A
\\ & \Rightarrow \det{AE}=(\det E)(\det A)
\end{align*}
If $S$ is equivalent to one row interchange, then
\begin{align*}
& \det E=-\det I = -1
\\ & \det{AE}=\det{(S(A))}=-\det A
\\ & \Rightarrow \det{AE} = (\det E)(\det A)
\end{align*}
If $S$ is equivalent to one row scaling of $k$, then
\begin{align*}
& \det E=k\det I = k
\\ & \det{AE}=\det{(S(A))}=k\det A
\\ & \Rightarrow \det{AE} = (\det E)(\det A)
\end{align*}