If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A} \mathbf{B}).$
We have that $\det (\mathbf{A} \mathbf{B}) = (\det \mathbf{A})(\det \mathbf{B}) = (2)(12) = \boxed{24}.$
24
Level 1
Precalculus
Problem: If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A} \mathbf{B}).$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the determinant of the product of two matrices, we use the property that states $\det(\mathbf{A} \mathbf{B}) = \det(\mathbf{A}) \cdot \det(\mathbf{B})$. This property holds for any square matrices $\mathbf{A}$ and $\mathbf{B}$. 

Given that $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12$, we can apply this property directly. 

First, we calculate the product of the determinants:
1. Compute $\det(\mathbf{A}) \cdot \det(\mathbf{B})$.
2. Substitute the known values: $2 \cdot 12$.
3. Perform the multiplication: $2 \cdot 12 = 24$.

Thus, the determinant of the product $\det(\mathbf{A} \mathbf{B})$ is $24$.<|END_OF_LATENT|> The final answer is $24$.