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?

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 $-4$. 
