To determine which matrix [tex]\( B \)[/tex] corresponds to the given condition [tex]\(|B| = -|A|\)[/tex] where [tex]\( A = \left[\begin{array}{cc}-2 & 5 \\ 7 & 6\end{array}\right] \)[/tex], let's proceed in a detailed and step-by-step manner.
### Step 1: Calculate the Determinant of Matrix [tex]\( A \)[/tex]
For matrix [tex]\( A = \left[\begin{array}{cc}-2 & 5 \\ 7 & 6\end{array}\right] \)[/tex],
the determinant [tex]\(|A|\)[/tex] is calculated as follows:
[tex]\[
|A| = (-2 \times 6) - (5 \times 7) = -12 - 35 = -47
\][/tex]
### Step 2: Determine the Required Determinant for Matrix [tex]\( B \)[/tex]
Given the condition [tex]\(|B| = -|A|\)[/tex],
[tex]\[
|B| = -(-47) = 47
\][/tex]
### Step 3: Calculate the Determinant of Each Possible Matrix [tex]\( B \)[/tex]
Option A: [tex]\( B_1 = \left[\begin{array}{cc}1 & -7 \\ 6 & 3\end{array}\right] \)[/tex]
[tex]\[
|B_1| = (1 \times 3) - (-7 \times 6) = 3 + 42 = 45
\][/tex]
Option B: [tex]\( B_2 = \left[\begin{array}{cc}9 & 2 \\ 3 & 6\end{array}\right] \)[/tex]
[tex]\[
|B_2| = (9 \times 6) - (2 \times 3) = 54 - 6 = 48
\][/tex]
Option C: [tex]\( B_3 = \left[\begin{array}{cc}12 & 1 \\ 13 & 5\end{array}\right] \)[/tex]
[tex]\[
|B_3| = (12 \times 5) - (1 \times 13) = 60 - 13 = 47
\][/tex]
Option D: [tex]\( B_4 = \left[\begin{array}{cc}4 & -3 \\ 6 & 7\end{array}\right] \)[/tex]
[tex]\[
|B_4| = (4 \times 7) - (-3 \times 6) = 28 + 18 = 46
\][/tex]
### Step 4: Identify the Correct Matrix [tex]\( B \)[/tex]
We are looking for the matrix whose determinant is [tex]\( 47 \)[/tex]. From our calculations, we see:
- [tex]\(|B_1| = 45\)[/tex]
- [tex]\(|B_2| = 48\)[/tex]
- [tex]\(|B_3| = 47\)[/tex]
- [tex]\(|B_4| = 46\)[/tex]
Thus, the matrix [tex]\( B_3 = \left[\begin{array}{cc}12 & 1 \\ 13 & 5\end{array}\right] \)[/tex] satisfies the condition [tex]\(|B| = 47\)[/tex].
Therefore, matrix [tex]\( B \)[/tex] is:
[tex]\[
\boxed{\left[\begin{array}{cc}12 & 1 \\ 13 & 5\end{array}\right]}
\][/tex]
