I want to compute the determinant of the matrix $ \left( \begin{matrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 6 \end{matrix} \right)\, $.
I use elimination in order to transform the matrix into a triangular one. I perform the following row operations:
- Swap $(row1)$ and $(row4)$ $$ \left( \begin{matrix} 1 & -4 & 0 & 6 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 2 & -8 & 6 & 8 \end{matrix} \right)\,. $$
- $-3(row1)+(row2)$, $3(row1)+(row3)$ and $-2(row1)+(row4)$ $$ \left( \begin{matrix} 1 & -4 & 0 & 6 \\ 0 & 3 & 5 & -8 \\ 0 & -12 & 1 & 16 \\ 0 & 0 & 6 & -4 \end{matrix} \right)\,. $$
- $4(row2)+(row3)$ $$ \left( \begin{matrix} 1 & -4 & 0 & 6 \\ 0 & 3 & 5 & -8 \\ 0 & 0 & 21 & -16 \\ 0 & 0 & 6 & -4 \end{matrix} \right)\,. $$
- Swap $(row3)$ and $(row4)$ $$ \left( \begin{matrix} 1 & -4 & 0 & 6 \\ 0 & 3 & 5 & -8 \\ 0 & 0 & 6 & -4 \\ 0 & 0 & 21 & -16 \end{matrix} \right)\,. $$
- $-{21\over 6}(row3)+(row4)$ $$ \left( \begin{matrix} 1 & -4 & 0 & 6 \\ 0 & 3 & 5 & -8 \\ 0 & 0 & 6 & -4 \\ 0 & 0 & 0 & -2 \end{matrix} \right)\,. $$
After elimination, since there are no zero rows, I multiply the pivots of each row (elements on the diagonal) and also multiply by $(-1)^{\text{number of row swaps}}$. The result following these steps is $-36$, but the actual result is $36$. Can anyone see the mistake?
EDIT: The actual determinant is $-36$. Probably I made a mistake when I checked the result in Mathematica.
Good news! The actual result is $-36$. :)