$\left( \begin{matrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\end{matrix} \right)=A$
Calculate Step 1:$\left( \begin{matrix} a_{11}&a_{12}\\ a_{21}&a_{22}\\ a_{31}&a_{32}\\ a_{41}&a_{42}\end{matrix} \right)$ and $\left( \begin{matrix} a_{13}&a_{14}\\ a_{23}&a_{24}\\ a_{33}&a_{34}\\ a_{43}&a_{44}\end{matrix} \right)$
Step 2: Each column is multiplied with the other column which is under. We will plus all results but we have to put negative(-) on second and fifth column For example: $\left( \begin{matrix} 1&2&3&0\\ 2&0&1&3\\ 1&1&0&1\\ 0&2&3&1\end{matrix} \right)$
How do I prove this method? Note: This rule available only $4\times 4$ determinant
If you define the determinant of a matrix to be:
$$\det \left( \begin{matrix} {a}&{b}&{c}&{d}\\{e}&{f}&{g}&{h}\\{i}&{j}&{k}&{l}\\{l}&{m}&{n}&{o}\end{matrix} \right) = a \det \left( \begin{matrix} {f}&{g}&{h}\\{j}&{k}&{l}\\{m}&{n}&{o}\end{matrix} \right) - b \det \left( \begin{matrix} {e}&{g}&{h}\\{i}&{k}&{l}\\{l}&{n}&{o}\end{matrix} \right)+c \det \left( \begin{matrix} {e}&{f}&{h}\\{i}&{j}&{l}\\{l}&{m}&{o}\end{matrix} \right)- d \det \left( \begin{matrix} {e}&{f}&{g}\\{i}&{j}&{k}\\{l}&{m}&{n}\end{matrix} \right)$$
http://mathworld.wolfram.com/Determinant.html
Then you can use your method with those 16 variables to show they are equal.