To determine whether matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are inverse matrices, we need to verify if their product, [tex]\(AB\)[/tex], is the identity matrix [tex]\(I\)[/tex]. The identity matrix for a 2x2 matrix is:
[tex]\[ I = \left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right] \][/tex]
Given the matrices:
[tex]\[ A = \left[\begin{array}{cc}
-6 & 4 \\
4 & -2
\end{array}\right] \][/tex]
[tex]\[ B = \left[\begin{array}{cc}
\frac{1}{2} & 1 \\
1 & \frac{3}{2}
\end{array}\right] \][/tex]
Let's multiply these matrices to check if their product is the identity matrix.
1. Compute the element in the first row and first column of [tex]\(AB\)[/tex]:
[tex]\[
(-6 \cdot \frac{1}{2}) + (4 \cdot 1) = -3 + 4 = 1
\][/tex]
2. Compute the element in the first row and second column of [tex]\(AB\)[/tex]:
[tex]\[
(-6 \cdot 1) + (4 \cdot \frac{3}{2}) = -6 + 6 = 0
\][/tex]
3. Compute the element in the second row and first column of [tex]\(AB\)[/tex]:
[tex]\[
(4 \cdot \frac{1}{2}) + (-2 \cdot 1) = 2 - 2 = 0
\][/tex]
4. Compute the element in the second row and second column of [tex]\(AB\)[/tex]:
[tex]\[
(4 \cdot 1) + (-2 \cdot \frac{3}{2}) = 4 - 3 = 1
\][/tex]
Thus, the product [tex]\(AB\)[/tex] is:
[tex]\[
AB = \left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]
\][/tex]
Since [tex]\(AB\)[/tex] is the identity matrix, matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are indeed inverses of each other. Therefore, the statement is:
True
