To solve for the determinant of $A \times B$, where:

$A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} -2 & 5 \\ 2 & -3 \end{pmatrix}$

The property of determinants states that:

$\text{det}(A \times B) = \text{det}(A) \times \text{det}(B)$

Step 1: Find the determinant of matrix $A$

$\text{det}(A) = (2 \times 3) - (5 \times 1) = 6 - 5 = 1$

Step 2: Find the determinant of matrix $B$

$\text{det}(B) = (-2 \times -3) - (5 \times 2) = 6 - 10 = -4$

Step 3: Calculate the determinant of $A \times B$

$\text{det}(A \times B) = \text{det}(A) \times \text{det}(B) = 1 \times (-4) = -4$

Therefore, the determinant of $A \times B$ is $-4$.

The correct answer is d. -4.

Would you like more details or have any questions? Here are some related questions for further practice:

1. How do you compute the determinant of a 3x3 matrix?
2. What is the determinant of a diagonal matrix?
3. Can a matrix with a zero determinant be inverted?
4. How does the determinant change when two rows of a matrix are swapped?
5. What are the properties of the determinant in relation to matrix multiplication?

Tip: The determinant of a product of two matrices is always the product of their individual determinants.