[SOLVED] Triangular matrix/determinant

The problem says "Reduce the following matrix into an upper triangular one in order to find its determinant".

I wonder why they ask me to reduce it since the calculus would only get more complicated.

So here is the matrix $\displaystyle \left( \begin{array}{ccc}\\ -1 & 3 & 1 \\ 2 & 5 & 3 \\ 1 & -2 & 1 \end{array}\right)$ I call it $\displaystyle A$. I reduced it to this one : $\displaystyle \left( \begin{array}{ccc}\\ 1 & -2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right)$ I call it $\displaystyle Z$. I calculate the determinant of $\displaystyle Z$ which is $\displaystyle 1$. By curiosity I calculated the one from $\displaystyle A$ and surprisingly I got $\displaystyle -17$... So my guess is that I made an error when I reducted $\displaystyle A$ into $\displaystyle Z$. Am I right saying this? Because I think the determinant of $\displaystyle A$ and $\displaystyle Z$ must be the same, right?

Last question : why do they wanted me to reduce $\displaystyle A$? It's much more simple to directly calculate $\displaystyle \det (A)$ without passing by a reduction of matrix and then calculating $\displaystyle \det (Z)$.