Let \(F\) be a field and let \(A\in M_{n\times n}(F)\) be a square matrix. For \(i\in \{1,\ldots ,n\}\) let \(A_{i}\in M_{(n-1)\times (n-1)}(F)\) be a square matrix gained by deleting the first column and the \(i\)-th row of \(A\).
The determinant \(|A|\) is defined recursively by \[|M|={\begin{cases}a_{11}\,,&{\text{if }}n=1\,,\\\sum _{i=1}^{n}(-1)^{i+1}a_{i1}|A_{i}|&{\text{ for }}n\geq 2\,.\end{cases}}\]
\[\begin{array}{rcl} \left|\begin{array}{cccc}a&b&c&d\\e&f&g&h\\i&j&k&l\\m&n&o&p\end{array}\right|&=&a\left|\begin{array}{ccc}f&g&h\\j&k&l\\n&o&p\end{array}\right|-e\left|\begin{array}{ccc}b&c&d\\j&k&l\\n&o&p\end{array}\right|+i\left|\begin{array}{ccc}b&c&d\\f&g&h\\n&o&p\end{array}\right|-m\left|\begin{array}{ccc}b&c&d\\f&g&h\\j&k&l\end{array}\right|\\ &=&af\left|\begin{array}{cc}k&l\\o&p\end{array}\right|-aj\left|\begin{array}{cc}g&h\\o&p\end{array}\right|+an\left|\begin{array}{cc}g&h\\k&l\end{array}\right|- eb\left|\begin{array}{cc}k&l\\o&p\end{array}\right|+ej\left|\begin{array}{cc}c&d\\o&p\end{array}\right|-en\left|\begin{array}{cc}c&d\\k&l\end{array}\right|+\\ &&ib\left|\begin{array}{cc}g&h\\o&p\end{array}\right|-if\left|\begin{array}{cc}c&d\\o&p\end{array}\right|+in\left|\begin{array}{cc}c&d\\g&h\end{array}\right|- mb\left|\begin{array}{cc}g&h\\k&l\end{array}\right|+mf\left|\begin{array}{cc}c&d\\k&l\end{array}\right|-mj\left|\begin{array}{cc}c&d\\g&h\end{array}\right|\\ &=&afkp-afol-ajgp+ajoh+angl-ankh-ebkp+ebol+ejcp-ejod-encl+enkd+\\ &&ibgp-iboh-ifcp+ifod+inch-ingd-mbgl+mbkh+mfcl-mfkd-mjch+mjgd \end{array}\]
