# Definition: Recursive Definition of the Determinant

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 $|M|={\begin{cases}a_{11}\,,&{\text{if }}n=1\,,\\\sum _{i=1}^{n}(-1)^{i+1}a_{i1}|A_{i}|&{\text{ for }}n\geq 2\,.\end{cases}}$

### Example - Calculating the Determinant of a (4x4)-Matrix

$\begin{array}{rcl} \left|\begin{array}{cccc}a&b&c&d\\e&f&g&h\\i&j&k&l\\m&n&o&p\end{array}\right|&=&a\left|\begin{array}{ccc}f&g&h\\j&k&l\\n&o&p\end{array}\right|-e\left|\begin{array}{ccc}b&c&d\\j&k&l\\n&o&p\end{array}\right|+i\left|\begin{array}{ccc}b&c&d\\f&g&h\\n&o&p\end{array}\right|-m\left|\begin{array}{ccc}b&c&d\\f&g&h\\j&k&l\end{array}\right|\\ &=&af\left|\begin{array}{cc}k&l\\o&p\end{array}\right|-aj\left|\begin{array}{cc}g&h\\o&p\end{array}\right|+an\left|\begin{array}{cc}g&h\\k&l\end{array}\right|- eb\left|\begin{array}{cc}k&l\\o&p\end{array}\right|+ej\left|\begin{array}{cc}c&d\\o&p\end{array}\right|-en\left|\begin{array}{cc}c&d\\k&l\end{array}\right|+\\ &&ib\left|\begin{array}{cc}g&h\\o&p\end{array}\right|-if\left|\begin{array}{cc}c&d\\o&p\end{array}\right|+in\left|\begin{array}{cc}c&d\\g&h\end{array}\right|- mb\left|\begin{array}{cc}g&h\\k&l\end{array}\right|+mf\left|\begin{array}{cc}c&d\\k&l\end{array}\right|-mj\left|\begin{array}{cc}c&d\\g&h\end{array}\right|\\ &=&afkp-afol-ajgp+ajoh+angl-ankh-ebkp+ebol+ejcp-ejod-encl+enkd+\\ &&ibgp-iboh-ifcp+ifod+inch-ingd-mbgl+mbkh+mfcl-mfkd-mjch+mjgd \end{array}$

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck