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
On
If you want to do it purely by row reduction:
First add 6 times the fourth row to the first one, we get \begin{align} \begin{vmatrix} 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{vmatrix} =\begin{vmatrix} 0& 0& −68& −13& 11\\ 0& 0& 0& −9& −7\\ 0& 15& 35& 0& 0\\ 0 &−1 &−11& −2& 1\\ −2 &−2& 3& 0& −2\end{vmatrix}. \end{align} Now add 15 times the fourth row to the third one: $$ \begin{vmatrix} 0& 0& −68& −13& 11\\ 0& 0& 0& −9& −7\\ 0& 0& -130& -30& 15\\ 0 &−1 &−11& −2& 1\\ −2 &−2& 3& 0& −2\end{vmatrix}. $$ Now multiply the first row by 65 and the third one by 34 (of, course, factoring those number out as divisors: $$ \frac1{34\times65}\,\begin{vmatrix} 0& 0& -4420& -845& 715\\ 0& 0& 0& −9& −7\\ 0& 0& -4420& -1020& 510\\ 0 &−1 &−11& −2& 1\\ −2 &−2& 3& 0& −2\end{vmatrix}. $$ Now subtract the third row from the first one: $$ \frac1{34\times65}\,\begin{vmatrix} 0& 0& 0& 175& 205\\ 0& 0& 0& −9& −7\\ 0& 0& -4420& -1020& 510\\ 0 &−1 &−11& −2& 1\\ −2 &−2& 3& 0& −2\end{vmatrix}. $$ Now multiply the first row by 9 and the second one by 175: $$\frac1{9\times34\times65\times175}\,\begin{vmatrix} 0& 0& 0& 1575& 1845\\ 0& 0& 0& −1575& −1225\\ 0& 0& -4420& -1020& 510\\ 0 &−1 &−11& −2& 1\\ −2 &−2& 3& 0& −2\end{vmatrix} $$ and then add the second row to the first one: $$ \frac1{9\times34\times65\times175}\,\begin{vmatrix} 0& 0& 0& 0& 620\\ 0& 0& 0& −1575& −12255\\ 0& 0& -4420& -1020& 510\\ 0 &−1 &−11& −2& 1\\ −2 &−2& 3& 0& −2\end{vmatrix} $$
The determinant is then $$ \frac{(-2)\times1\times (-4420)\times1575\times620}{9\times34\times65\times175}=2480. $$
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 $$ R = 10\left[-9\det\begin{pmatrix}6&-2& 5 \\ 3 & 7 & 0 \\ -1&-11&1\end{pmatrix}-7\det\begin{pmatrix}6&-2&1 \\ 3 & 7 & 0 \\ -1&-11&2\end{pmatrix}\right]$$ $$ R = 10\left[-9\det\begin{pmatrix}11&53& 0 \\ 3 & 7 & 0 \\ -1&-11&1\end{pmatrix}-7\det\begin{pmatrix}6&-2&1 \\ 3 & 7 & 0 \\ -13&-7&0\end{pmatrix}\right]$$ $$ R = 10\left[-9\cdot(11\cdot 7-53\cdot 3)-7\cdot\left(-7\cdot 3+7\cdot 13\right)\right]=\color{red}{2480}.$$