Determinante

Eine determinante kann anzeigen, ob eine Matrix invertierbar ist.

Berechnung der Determinante

Definition Determinante 2x2 Matrix

$$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

$$\det (A)=\mid A\mid =a\ast d-b\ast c$$

## Definition Determinante 3x3 Matrix

$$A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$$

$$\begin{gathered} \det (A)=\mid A\mid \\ =a_{11}\cdot a_{22}\cdot a_{33}+a_{12}\cdot a_{23}\cdot a_{31}+a_{13}\cdot a_{21}\cdot a_{32} \\ -a_{13}\cdot a_{22}\cdot a_{31}-a_{11}\cdot a_{23}\cdot a_{32}-a_{12}\cdot a_{21}\cdot a_{33} \end{gathered}$$

### Geometrische Interpretation

$$\mid \left(\begin{array}{c}{a}_1\\ a_2\\ 0\end{array}\right)\times \left(\begin{array}{c}{b}_1\\ b_2\\ 0\end{array}\right)\mid =\mid \left(\begin{array}{c}0\\ 0\\ a_1\cdot b_2-a_2\cdot b_1\end{array}\right)\mid =\mid a_1\cdot b_2-a_2\cdot b_1\mid$$

$$\det (A\cdot B)=\det (A)\cdot \det (B)$$

$$\det (A^{-1})=\frac{1}{\det (A)}$$

$$\det (\lambda \cdot A)={\lambda }^n\cdot \det (A)$$

$$\det (E)=1$$