Key Concept: Properties of Determinants and Algebraic Comparison

This problem requires us to evaluate a $3 \times 3$ determinant using elementary row and column operations. The goal is to simplify the determinant into a factored form and then compare it with a given algebraic expression to determine the values of constants $A$ and $B$. The key properties of determinants we will use are:

1. Column/Row Operations: Adding a multiple of one column (or row) to another column (or row) does not change the value of the determinant. This is useful for creating zeros or common factors.
2. Factoring: A common factor from any single row or column can be taken out of the determinant.
3. Triangular Determinant: The determinant of a triangular