<s>[INST] Finding determinants using both reduction and cofactor expansion I know that you can find the determinant of a matrix by either row reducing so that it is upper triangular and then multiplying the diagonal entries, or by expanding by cofactors. But could I reduce the matrix halfway (not entirely reduced to the point where it is in upper triangular) and then do cofactor expansion? Would that give me the same determinant?
 [/INST] Yes, provided you keep track of the changes to the determinant. Any combination of row reductions and cofactor expansions can be used. For example
$$\begin{vmatrix}5 & 2 & 3 \\ 12 & 4 & 6 \\ 3 & 4 & 7\end{vmatrix} = 2\begin{vmatrix}5 & 2 & 3 \\ 6 &2 & 3 \\ 3 & 4 & 7\end{vmatrix} = 2\begin{vmatrix}5 & 2 & 3 \\ 1 & 0 & 0 \\ 3 & 4 & 7\end{vmatrix}$$
Where we have first factored out a $2$ from row $2$ and then subtracted row $1$ from row $2$. Now we expand along row $2$ to get
$$2\begin{vmatrix}5 & 2 & 3 \\ 1 &0 & 0 \\ 3 & 4 & 7\end{vmatrix} = 2(-1)^{2+1}\begin{vmatrix} 2 & 3 \\ 4 & 7\end{vmatrix} = -2\begin{vmatrix} 2 & 3 \\ 0 & 1\end{vmatrix}$$
where in the last step we subtract twice row $1$ from row $2$. Now we simply multiply the diagonal entries to get determinant equal to $-4$. 
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