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

De rekenmachine berekent de determinant van de vierkante $$$2$$$x$$$2$$$-matrix $$$\left[\begin{array}{cc}- r \sin{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \cos{\left(\tanh{\left(\eta \right)} \right)}\\r \cos{\left(\tanh{\left(\eta \right)} \right)} \operatorname{sech}^{2}{\left(\eta \right)} & \sin{\left(\tanh{\left(\eta \right)} \right)}\end{array}\right]$$$, waarbij de stappen worden getoond.

Gerelateerde rekenmachine: Rekenmachine voor cofactormatrix

A

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Uw invoer

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

Oplossing

De determinant van een 2x2-matrix is $$$\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$$$.

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

Antwoord

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