The problem asks for the determinant of matrix $B$, which is derived from matrix $A$ by multiplying the first row of $A$ by $\frac{1}{2}$. We are also given that the determinant of $A$, denoted as $\text{det}(A)$, is 12.

Key concept:

When a row of a matrix is multiplied by a scalar, the determinant of the new matrix is multiplied by that scalar. Specifically, if a row of a matrix $A$ is multiplied by a scalar $k$, then the determinant of the new matrix is: $\text{det}(B) = k \cdot \text{det}(A)$ Here, the first row of matrix $A$ is multiplied by $\frac{1}{2}$, so the determinant of $B$ is: $\text{det}(B) = \frac{1}{2} \cdot \text{det}(A)$ Substituting the given value $\text{det}(A) = 12$: $\text{det}(B) = \frac{1}{2} \cdot 12 = 6$

Thus, the determinant of matrix $B$ is $6$.

Let me know if you need further details or clarifications!

Related questions:

1. What happens to the determinant if the second row of a matrix is multiplied by $\frac{1}{3}$?
2. How would the determinant change if all rows of a matrix are multiplied by a constant?
3. If two rows of a matrix are swapped, how does that affect the determinant?
4. Can you explain how row operations affect the determinant?
5. What happens to the determinant of a matrix if one row is added to another row?

Tip:

Always remember that multiplying a row by a scalar multiplies the determinant by that scalar. This rule only applies to single rows, not the entire matrix.