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Determinante von $$$\left[\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right]$$$

Der Rechner berechnet die Determinante der quadratischen $$$2$$$x$$$2$$$-Matrix $$$\left[\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right]$$$, wobei die Rechenschritte angezeigt werden.

Verwandter Rechner: Kofaktormatrix-Rechner

A

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Ihre Eingabe

Berechne $$$\left|\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right|.$$$

Lösung

Die Determinante einer 2x2-Matrix ist $$$\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$$$.

$$$\left|\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right| = \left(- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)}\right)\cdot \left(\sin{\left(\tanh{\left(\eta \right)} \right)}\right) - \left(\cos{\left(\tanh{\left(\eta \right)} \right)}\right)\cdot \left(r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)}\right) = - r \operatorname{sech}^{2}{\left(\eta \right)}$$$

Antwort

$$$\left|\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right| = - r \operatorname{sech}^{2}{\left(\eta \right)}$$$A