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The image contains the following mathematical problem written in Indonesian:

We are given two matrices:

\[
P = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}, \quad Q = \begin{pmatrix} 5 & 4 \\ 1 & 1 \end{pmatrix}
\]

The text further says:

- If \( P^{-1} \) is the inverse of matrix \( P \) and \( Q^{-1} \) is the inverse of matrix \( Q \),
- Then, what is the determinant of \( (PQ)^{-1} \)?

### Solution Approach:

1. **Matrix Multiplication Property**: 
   The determinant of the inverse of a matrix \( A \) is given by:
   \[
   \text{det}(A^{-1}) = \frac{1}{\text{det}(A)}
   \]
   Therefore:
   \[
   \text{det}((PQ)^{-1}) = \frac{1}{\text{det}(PQ)}
   \]
   
2. **Determinant of Product of Matrices**: 
   The determinant of a product of two matrices is the product of their determinants:
   \[
   \text{det}(PQ) = \text{det}(P) \times \text{det}(Q)
   \]
   So,
   \[
   \text{det}((PQ)^{-1}) = \frac{1}{\text{det}(P) \times \text{det}(Q)}
   \]

3. **Finding Determinants**:
   For matrix \( P \):
   \[
   \text{det}(P) = (2 \times 3) - (5 \times 1) = 6 - 5 = 1
   \]
   For matrix \( Q \):
   \[
   \text{det}(Q) = (5 \times 1) - (4 \times 1) = 5 - 4 = 1
   \]

4. **Final Calculation**:
   \[
   \text{det}((PQ)^{-1}) = \frac{1}{1 \times 1} = 1
   \]

### Conclusion:
The determinant of \( (PQ)^{-1} \) is **1**.

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

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Here are five related questions:
1. How do you find the determinant of a 3x3 matrix?
2. What are the properties of matrix inverses?
3. Can the determinant of a matrix ever be zero? What does that imply?
4. How does the inverse of a matrix relate to solving systems of linear equations?
5. What are the conditions for a matrix to have an inverse?

**Tip**: The determinant of a matrix can be thought of as a scalar that provides information about the matrix's behavior, such as whether it's invertible (non-zero determinant).

Diketahui matriks P = [[2, 5], [1, 3]] dan Q = [[5, 4], [1, 1]]. Jika P^-1 adalah invers matriks P dan Q^-1 adalah invers matriks Q, maka determinan matriks (PQ)^-1 adalah ...

This problem involves calculating the determinant of (PQ)^-1 using matrix properties. It explores matrix multiplication, determinants, and inverses with the matrices P and Q. Suitable for Grades 10-12.

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