To solve the problem, we need to determine which matrix [tex]\( B \)[/tex] gives the correct determinant relationship with matrix [tex]\( A \)[/tex].
1. Calculate the determinant of [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
-2 & 5 \\
7 & 6
\end{pmatrix}
\][/tex]
The determinant of [tex]\( A \)[/tex] is calculated as follows:
[tex]\[
\text{det}(A) = (-2 \cdot 6) - (5 \cdot 7) = -12 - 35 = -47
\][/tex]
Therefore, [tex]\(|A| = -47\)[/tex].
2. Determine the target determinant for [tex]\( B \)[/tex]:
Given that [tex]\(|B| = -|A|\)[/tex], we have:
[tex]\[
|B| = -(-47) = 47
\][/tex]
3. Check which matrix [tex]\( B \)[/tex] has a determinant of 47:
- Option A:
[tex]\[
B = \begin{pmatrix}
1 & -7 \\
6 & 3
\end{pmatrix}
\][/tex]
[tex]\[
\text{det}(B) = (1 \cdot 3) - (-7 \cdot 6) = 3 + 42 = 45
\][/tex]
This is incorrect since [tex]\( \text{det}(B) \neq 47 \)[/tex].
- Option B:
[tex]\[
B = \begin{pmatrix}
9 & 2 \\
3 & 6
\end{pmatrix}
\][/tex]
[tex]\[
\text{det}(B) = (9 \cdot 6) - (2 \cdot 3) = 54 - 6 = 48
\][/tex]
This is incorrect since [tex]\( \text{det}(B) \neq 47 \)[/tex].
- Option C:
[tex]\[
B = \begin{pmatrix}
12 & 1 \\
13 & 5
\end{pmatrix}
\][/tex]
[tex]\[
\text{det}(B) = (12 \cdot 5) - (1 \cdot 13) = 60 - 13 = 47
\][/tex]
This is correct since [tex]\( \text{det}(B) = 47 \)[/tex].
- Option D:
[tex]\[
B = \begin{pmatrix}
4 & -3 \\
6 & 7
\end{pmatrix}
\][/tex]
[tex]\[
\text{det}(B) = (4 \cdot 7) - (-3 \cdot 6) = 28 + 18 = 46
\][/tex]
This is incorrect since [tex]\( \text{det}(B) \neq 47 \)[/tex].
Therefore, the correct matrix [tex]\( B \)[/tex] is:
[tex]\[
\boxed{\begin{pmatrix} 12 & 1 \\ 13 & 5 \end{pmatrix}}
\][/tex]
