Reducing a matrix to reduced row echelon form without introducing fractions I've been trying to figure out how to reduce this matrix without introducing fractions in the intermediate stages but can't figure out how to do it.
$$\begin{bmatrix}2&1&3\\
0&-2&-29\\
3&4&5\end{bmatrix}$$

Here's one way.
$\begin{bmatrix}2&1&3\\
0&-2&-29\\
3&4&5\end{bmatrix}
\xrightarrow  {R3-R1}
\begin{bmatrix}2&1&3\\
0&-2&-29\\
1&3&2
\end{bmatrix}
\xrightarrow  {R1-2R3}
\begin{bmatrix}0&-5&-1\\
0&-2&-29\\
1&3&2
\end{bmatrix}
\xrightarrow  {-R1}
\begin{bmatrix}0&5&1\\
0&-2&-29\\
1&3&2
\end{bmatrix}
\xrightarrow  {R1+2R2}
\begin{bmatrix}0&1&-57\\
0&-2&-29\\
1&3&2
\end{bmatrix}
\xrightarrow  {R2+2R1}
\begin{bmatrix}0&1&-57\\
0&0&-143\\
1&3&2
\end{bmatrix}
\xrightarrow{-R2/143}
\begin{bmatrix}0&1&-57\\
0&0&1\\
1&3&2
\end{bmatrix}
\xrightarrow{R1+57R2}
\begin{bmatrix}0&1&0\\
0&0&1\\
1&3&2
\end{bmatrix}
\xrightarrow[R3-2R2]{R3-3R1}
\begin{bmatrix}0&1&0\\
0&0&1\\
1&0&0
\end{bmatrix}
\xrightarrow[R2\leftrightarrow R3]{R2\leftrightarrow R3}
\begin{bmatrix}1&0&0\\
0&1&0\\
0&0&1
\end{bmatrix}
$
