In the provided image, you are asked to verify the identity:

$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$

for two given matrices $A$ and $B$:

$A = \begin{pmatrix} 4 & -1 \\ 2 & -0.125 \end{pmatrix}, \quad B = \begin{pmatrix} -0.5 & 2 \\ 4 & 8 \end{pmatrix}.$

Steps to solve:

1. Calculate $\text{det}(A)$: The determinant of a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by:

   $\text{det}(A) = ad - bc$

2. Calculate $\text{det}(B)$: Use the same formula for matrix $B$.

3. Multiply matrices $A$ and $B$: Perform matrix multiplication to compute $AB$.

4. Calculate $\text{det}(AB)$: After finding $AB$, compute its determinant.

5. Compare: Check if $\text{det}(AB)$ is equal to $\text{det}(A) \cdot \text{det}(B)$.

Let me calculate these values for you.Here are the results:

1. $\text{det}(A) = 1.5$
2. $\text{det}(B) = -12.0$
3. The product $\text{det}(A) \cdot \text{det}(B) = 1.5 \times -12.0 = -18.0$
4. $\text{det}(AB) = -18.0$ (approximately $-17.999999999999996$ due to rounding).

Thus, the identity $\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$ holds true for the given matrices.

Would you like a deeper explanation of the steps?

Here are 5 related questions:

1. How is the determinant of a 2x2 matrix derived?
2. What happens when one of the matrices has a zero determinant?
3. Can this identity be extended to larger matrices?
4. How would rounding affect larger matrices in determinant calculations?
5. What other matrix identities are important to know?

Tip: When working with floating-point numbers, slight differences in results can occur due to rounding errors, especially in large calculations.