How to find the determinant of a 5x5 matrix How do I find the determinant of this?
$$\begin{bmatrix}
0& 6& −2& −1& 5\\
0& 0& 0& −9& −7\\
0& 15& 35& 0& 0\\
0 &−1 &−11& −2& 1\\
−2 &−2& 3& 0& −2\end{bmatrix}$$
I tried doing row reductions but every time I get a $0$ and I gain a number. I'm not really sure how to do this by cofactors

By using a Laplace expansion along the first column the problem immediately boils down to computing $R=-2\cdot\det(M)$ with
$$ \det M=\det\begin{pmatrix}6&-2&-1& 5 \\ 0 & 0 & -9 & -7 \\ 15 & 35 & 0 & 0 \\ -1&-11&-2&1\end{pmatrix}=-5\cdot\det\begin{pmatrix}6&-2&1& 5 \\ 0 & 0 & 9 & -7 \\ 3 & 7 & 0 & 0 \\ -1&-11&2&1\end{pmatrix}$$
hence
$$ R = 10\left[-9\det\begin{pmatrix}6&-2& 5 \\ 3 & 7  & 0 \\ -1&-11&1\end{pmatrix}-7\det\begin{pmatrix}6&-2&1 \\ 3 & 7 & 0  \\ -1&-11&2\end{pmatrix}\right]$$
$$ R = 10\left[-9\det\begin{pmatrix}11&53& 0 \\ 3 & 7  & 0 \\ -1&-11&1\end{pmatrix}-7\det\begin{pmatrix}6&-2&1 \\ 3 & 7 & 0  \\ -13&-7&0\end{pmatrix}\right]$$
$$ R = 10\left[-9\cdot(11\cdot 7-53\cdot 3)-7\cdot\left(-7\cdot 3+7\cdot 13\right)\right]=\color{red}{2480}.$$
