The problem asks to calculate the determinant of the matrix expression $\det(A^2 B^T A^{-1} B)$, given that:

• $\det(A) = 2$
• $\det(B) = 3$

Step-by-step solution:

1. Use the determinant properties:

   • $\det(A^2) = \det(A)^2$
   • $\det(B^T) = \det(B)$ because the determinant of a transpose is the same as the determinant of the original matrix.
   • $\det(A^{-1}) = \frac{1}{\det(A)}$

2. Simplify the determinant expression: $\det(A^2 B^T A^{-1} B) = \det(A^2) \cdot \det(B^T) \cdot \det(A^{-1}) \cdot \det(B)$

   Using the given values: $= \det(A)^2 \cdot \det(B) \cdot \frac{1}{\det(A)} \cdot \det(B)$ Substituting $\det(A) = 2$ and $\det(B) = 3$: $= 2^2 \cdot 3 \cdot \frac{1}{2} \cdot 3$ $= 4 \cdot 3 \cdot \frac{1}{2} \cdot 3$ $= 6 \cdot 9 = 54$

Final answer:

$\det(A^2 B^T A^{-1} B) = 54$

Would you like more details on any step? Here are some related questions to explore:

1. What are the key properties of determinants that simplify matrix operations?
2. How does the determinant of a transpose affect other matrix operations?
3. Why is the determinant of an inverse the reciprocal of the original matrix's determinant?
4. How do matrix exponentiation and determinants relate?
5. Can the determinant provide insights into the invertibility of a matrix?

Tip: When working with determinants, always check for opportunities to apply basic properties such as multiplicativity and the behavior of transposes and inverses!