This problem tests your understanding of properties of determinants, especially those related to scalar multiplication of matrices and adjugate matrices. We will systematically break down the problem using these properties.

1. Key Concepts and Formulas

For an $n \times n$ matrix $M$:

• Determinant of a scalar multiple: $|kM| = k^n |M|$, where $k$ is a scalar.
• Determinant of the adjugate: $|\operatorname{adj}(M)| = |M|^{n-1}$.
• Determinant of the adjugate of adjugate: $|\operatorname{adj}(\operatorname{adj}(M))| = |M|^{(n-1)^2}$.

In this problem, the matrix $A$