<s>[INST] Row Reduction with Cofactor Expansion My calculator says the determinant of $$\begin{pmatrix}3 &0&6&-3\\0&2&3&0\\-4&-7&2&0\\2&0&1&10\end{pmatrix}$$ is $396$. However, the website I got the matrix from says a number over 1,000, and my own answer is 12. I'm wondering how to correctly arrive at the answer 396. Can a good linear algebraist help?
 [/INST] $$\begin{pmatrix}3 &0&6&\!\!\!\!-3\\0&2&3&0\\\!\!\!\!-4&\!\!\!\!-7&2&0\\2&0&1&10\end{pmatrix}\stackrel{\text{R_4}+\frac{10}{3}\,\text{R_1}}\longrightarrow\begin{pmatrix}3 &0&6&\!\!\!-3\\0&2&3&0\\\!\!-4&\!\!-7&2&0\\12&0&21&0\end{pmatrix}$$
Develop the determinant wrt the fourth column, and I get:
$$\det A=3\,\,\left|\begin{pmatrix}0&2&3\\\!\!-4&\!\!-7&2\\12&0&21\end{pmatrix}\right|=3\left(48+168+252\right)=1,404$$
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