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?
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$\begingroup$ Not necessarily - performing the row operation of multiplying a row by a number other than 1 changes the determinant, as does switching two rows. But the gist of your idea is right. If you keep track of how the row operations change the determinant as you row reduce it to the point that you want to switch to the cofactor expansion then you can combine this with the result of doing the cofactor expansion to find the determinant of the original matrix. $\endgroup$ – Jonah Sinick Oct 31 '12 at 5:49
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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 $-4$.
