From Mathematical Methods for Engineers and Scientists 1 by K.T. Tang:
Example 4.6.1. Evaluate $$D_4=\begin{vmatrix}2&1&3&1\\1&0&2&5\\2&1&1&3\\1&3&0&2\end{vmatrix}$$ by (a) expansion with minors formed from the first two columns, (b) expansion with minors formed from the second and fourth rows.
Solution 4.6.1. (a) $$\begin{align}D_4&=\begin{vmatrix}2&1\\1&0\end{vmatrix}\cdot\begin{vmatrix}1&3\\0&2\end{vmatrix}-\begin{vmatrix}2&1\\2&1\end{vmatrix}\cdot\begin{vmatrix}2&5\\0&2\end{vmatrix}+\begin{vmatrix}2&1\\1&3\end{vmatrix}\cdot\begin{vmatrix}2&5\\1&3\end{vmatrix}\\&\phantom=\;+\begin{vmatrix}1&0\\2&1\end{vmatrix}\cdot\begin{vmatrix}3&1\\0&2\end{vmatrix}-\begin{vmatrix}1&0\\1&3\end{vmatrix}\cdot\begin{vmatrix}3&1\\1&3\end{vmatrix}+\begin{vmatrix}2&1\\1&3\end{vmatrix}\cdot\begin{vmatrix}3&1\\2&5\end{vmatrix}\\&=-2-0+5+6-24+65=50\end{align}$$ (b) $$\begin{align}D_4&=(-1)^{2+4+1+2}\begin{vmatrix}1&0\\1&3\end{vmatrix}\cdot\begin{vmatrix}3&1\\1&3\end{vmatrix}+(-1)^{2+4+1+3}\begin{vmatrix}1&2\\1&0\end{vmatrix}\cdot\begin{vmatrix}1&1\\1&3\end{vmatrix}\\&\phantom=\;+(-1)^{2+4+1+4}\begin{vmatrix}1&5\\1&2\end{vmatrix}\cdot\begin{vmatrix}1&3\\1&1\end{vmatrix}+(-1)^{2+4+2+3}\begin{vmatrix}0&2\\3&0\end{vmatrix}\cdot\begin{vmatrix}2&1\\2&3\end{vmatrix}\\&\phantom=\;+(-1)^{2+4+2+4}\begin{vmatrix}0&5\\3&2\end{vmatrix}\cdot\begin{vmatrix}2&4\\2&1\end{vmatrix}+(-1)^{2+4+3+4}\begin{vmatrix}2&5\\0&2\end{vmatrix}\cdot\begin{vmatrix}2&1\\2&1\end{vmatrix}\\&=-24-4-6+24+60-0=50\end{align}$$
Why did the author do part b differently than part a (aside from the obvious fact the the problem states to use rows instead of columns)? He multiplied each term by the cofactors. I don't really understand why he did that here and not in part a.