= Solution
Use $R_2\gets R_2+2R_1$ and $R_3\gets R_3-3R_1$, followed by $R_3\gets R_3+R_2$. These <elementary row operations> give
$$
\begin{pmatrix}1&4&1\\-2&-4&0\\3&8&0\end{pmatrix}\sim\begin{pmatrix}1&4&1\\0&4&2\\0&-4&-3\end{pmatrix}\sim\underbrace{\begin{pmatrix}1&4&1\\0&4&2\\0&0&-1\end{pmatrix}}_U.
$$
The elimination multipliers are $m_{21}=-2$, $m_{31}=3$, $m_{32}=-1$. Record them in the <unit lower triangular matrix> $L$:
$$
\boxed{A=LU,\qquad L=\begin{pmatrix}1&0&0\\-2&1&0\\3&-1&1\end{pmatrix},\quad U=\begin{pmatrix}1&4&1\\0&4&2\\0&0&-1\end{pmatrix}.}
$$
To see the <elementary matrix> relation directly, let $F_{21}=I+2e_{21}$, $F_{31}=I-3e_{31}$ and $F_{32}=I+e_{32}$, where $e_{ij}$ has a single 1 in position $(i,j)$. Then $U=F_{32}F_{31}F_{21}A$, and reversing and inverting these operations gives $L=F_{21}^{-1}F_{31}^{-1}F_{32}^{-1}$. Direct <matrix multiplication> also verifies the stated <LU decomposition>.
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