Explanation

(b)
$\left| A(\alpha,\beta) \right| = \cos^{2}{\alpha e^{\beta} + \sin^{2}{\alpha e^{\beta} = e^{\beta}}}$
Now,
$A(\alpha,\beta)^{- 1} = \frac{1}{e^{\beta}}adj\ (A(\alpha,\beta))$
$= \frac{1}{e^{\beta}}\begin{bmatrix}
e^{\beta}\cos\alpha & - \sin\alpha e^{\beta} & 0 \\
e^{\beta}\sin\alpha & \cos\alpha e^{\beta} & 0 \\
0 & 0 & 1
\end{bmatrix}$
$= \begin{bmatrix}
\cos\alpha & - \sin\alpha & 0 \\
\sin\alpha & \cos\alpha & 0 \\
0 & 0 & e^{- \beta}
\end{bmatrix}$
$= A( - \alpha, - \ \beta)$