Determinant

linear-algebra

Definition

Determinant

The determinant of a square matrix $A=(a_{ij})\in F^{n\times n}$ is given by:

$$\det A=\sum _{\pi \in S_n}\mathrm{sgn}(\pi )a_{\pi (1)1}{a}_{\pi (2)2}\dots a_{\pi (n)n}$$

where $S_n$ is the set of all permutations of $\{1,2,\dots ,n\}$ and $\mathrm{sgn}(\pi )$ is the signum of a permutation $\pi$.

For simplicity, the determinant can be writen as:

$$\det A=\left|\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \dots & a_{nn}\end{array}\right|$$

Properties

$$\begin{array}{rlrl}1.)& \left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|& & =1\\ 2.)& \left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|& & =-1\\ 3a.)& \left|\begin{array}{cc}ta& tb\\ c& d\end{array}\right|& & =t\cdot \left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ 3b.)& \left|\begin{array}{cc}a+a^{\prime }& b+b^{\prime }\\ c& d\end{array}\right|& & =\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|+\left|\begin{array}{cc}{a}^{\prime }& b^{\prime }\\ c& d\end{array}\right|\end{array}$$

¹

Rule of Sarrus

Definition

Rule of Sarrus

The number of permutations of a $3\times 3$ square matrix is $3!=6$. The rule of Sarrus states that the determinant of a $3\times 3$ square matrix is given by:

$$\begin{array}{rl}\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=& a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}a31+a_{13}{a}_{21}{a}_{32}\\ & -a_{13}{a}_{22}{a}_{31}-a_{12}{a}_{21}{a}_{33}-a_{11}{a}_{23}{a}_{32}\end{array}$$

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Footnotes

1. https://youtu.be/srxexLishgY?list=PL49CF3715CB9EF31D ↩