TITLE: Finding determinant by applying Gaussian Elimination
QUESTION [6 upvotes]: (I don't know how to make a matrix here, someone please correct it into a better format, thanks~)
So I'm applying the Gaussian Elimination to find the determinant for this matrix:
$\begin{pmatrix}
0&1&3\\
1&2&0\\
0&3&4\\
\end{pmatrix}$
So, I switched row $1$ and $2$: 
$\begin{pmatrix}
1&2&0\\
0&1&3\\
0&3&4\\
\end{pmatrix}$
Then, add the multiple of $-3$ of row $2$ to the third row: $\begin{pmatrix}
1&2&0\\
0&1&3\\
0&0&-5\\
\end{pmatrix}$
So the determinant I got is $-5$, however the answer key said it's $5$.  Some1 point out what I have done wrong?  Thank you!!

REPLY [7 votes]: When you apply Gaussian Elimination to find the determinant, you have to multiply by $(-1)$ at the end for every row switches you have applied.