If $\mathrm{A}=\left(\begin{array}{cc}\mathrm{cosh\theta }& \mathrm{sinh\theta }\\ \mathrm{sinh\theta }& \mathrm{cosh\theta }\end{array}\right) \mathrm{and} \mathrm{n}\in \mathrm{N}$ then ${\mathrm{A}}^{\mathrm{n}}=$

$$\begin{gathered} \mathrm{A}.\mathrm{A}=\left(\begin{array}{cc}\mathrm{cosh\theta }& \mathrm{sinh\theta }\\ \mathrm{sinh\theta }& \mathrm{cosh\theta }\end{array}\right)\left(\begin{array}{cc}\mathrm{cosh\theta }& \mathrm{sinh\theta }\\ \mathrm{sinh\theta }& \mathrm{cosh\theta }\end{array}\right) \\ {A}^2=\left(\begin{array}{cc}{\cosh }^2\theta +{\sinh }^2\theta & \mathrm{sinh\theta }\cos \mathrm{h\theta }+\mathrm{sinh\theta }\cos \mathrm{h\theta }\\ \mathrm{sinh\theta }\cos \mathrm{h\theta }+\mathrm{sinh\theta }\cos \mathrm{h\theta }& {\cosh }^2\theta +{\sinh }^2\theta \end{array}\right) \end{gathered}$$

$$\begin{gathered} {\mathrm{A}}^2=\left[\begin{array}{cc}\cos \mathrm{h}2\mathrm{\theta }& \sin \mathrm{h}2\mathrm{\theta }\\ \sin \mathrm{h}2\mathrm{\theta }& \cos \mathrm{h}2\mathrm{\theta }\end{array}\right] \\ \therefore {\mathrm{A}}^{\mathrm{n}}=\left[\begin{array}{cc}\cos \mathrm{h} \mathrm{n\theta }& \sin \mathrm{h} \mathrm{n\theta }\\ \sin \mathrm{h} \mathrm{n\theta }& \cos \mathrm{h} \mathrm{n\theta }\end{array}\right] \end{gathered}$$