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The problem states:

"Let \( A \) and \( B \) be two square matrices with \( \text{det}(A) = 2 \) and \( \text{det}(B) = 3 \). Calculate \( \text{det}(A^2 B^T A^{-1} B) \)."

### Solution:

We are given two square matrices \( A \) and \( B \), and we need to compute the determinant of the matrix expression \( A^2 B^T A^{-1} B \).

#### Step-by-step approach:
1. **Use the properties of determinants**: 
   - \( \text{det}(XY) = \text{det}(X) \cdot \text{det}(Y) \) for any matrices \( X \) and \( Y \) of the same size.
   - \( \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \).
   - \( \text{det}(A^T) = \text{det}(A) \) for any square matrix \( A \).

2. **Break down the expression \( A^2 B^T A^{-1} B \)**:
   - \( \text{det}(A^2) = \text{det}(A)^2 = 2^2 = 4 \).
   - \( \text{det}(B^T) = \text{det}(B) = 3 \).
   - \( \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} = \frac{1}{2} \).
   - \( \text{det}(B) = 3 \).

3. **Now calculate the determinant of the full expression**:
   \[
   \text{det}(A^2 B^T A^{-1} B) = \text{det}(A^2) \cdot \text{det}(B^T) \cdot \text{det}(A^{-1}) \cdot \text{det}(B)
   \]
   Substituting the values we computed:
   \[
   \text{det}(A^2 B^T A^{-1} B) = 4 \cdot 3 \cdot \frac{1}{2} \cdot 3 = 4 \cdot 9 = 36
   \]

Thus, the determinant of \( A^2 B^T A^{-1} B \) is \( \boxed{36} \).

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Let me know if you'd like more details or have any questions!

Here are five related questions:
1. What are some other properties of matrix determinants?
2. How does the determinant of a product of matrices relate to the individual determinants?
3. What is the significance of the determinant being non-zero for square matrices?
4. How does the transpose of a matrix affect its determinant?
5. How can we interpret determinants geometrically?

**Tip**: For inverse matrices, remember that \( \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \), which is useful in complex matrix determinant calculations.

Let A and B be two square matrices with det(A) = 2 and det(B) = 3. Calculate det(A^2 B^T A^{-1} B).

Learn how to calculate the determinant of the matrix expression A^2 B^T A^{-1} B, given det(A) = 2 and det(B) = 3. This problem covers key concepts like matrix transpose, inverse matrices, and determinant multiplication. Suitable for advanced high school or college students.

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