Step-by-Step Solution
Key Concept: Determinant properties: $|A^T| = |A|$, $|kA| = k^n|A|$, and the relationship between determinants of products
Given $A^T [A|B = A|B]B^T$, taking determinant on both sides: $|A^T||A|B| = |A|B|^T|$. Since $|A^T| = |A|$ and $|B^T| = |B|$, we get $|A| \cdot |A|B| = |A|B| \cdot |B|$. Using the property $|adj(A^T)| = |A|^{n-1}$ and solving the given matrix equation leads to $|A| = 2$.
Correct Answer: 2