Let's determine if the given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are inverses of one another by calculating the product [tex]\( AB \)[/tex].
Given matrices:
[tex]\[
A = \begin{pmatrix}
2 & 2 \\
1 & -1
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
\frac{1}{4} & \frac{1}{2} \\
\frac{1}{4} & -\frac{1}{2}
\end{pmatrix}
\][/tex]
First, we need to calculate the elements of the resulting matrix [tex]\( AB \)[/tex].
1. Calculate [tex]\( c_{11} \)[/tex]:
[tex]\[
c_{11} = (2 \cdot \frac{1}{4}) + (2 \cdot \frac{1}{4}) = \frac{2}{4} + \frac{2}{4} = \frac{4}{4} = 1
\][/tex]
2. Calculate [tex]\( c_{12} \)[/tex]:
[tex]\[
c_{12} = (2 \cdot \frac{1}{2}) + (2 \cdot -\frac{1}{2}) = 1 - 1 = 0
\][/tex]
3. Calculate [tex]\( c_{21} \)[/tex]:
[tex]\[
c_{21} = (1 \cdot \frac{1}{4}) + (-1 \cdot \frac{1}{4}) = \frac{1}{4} - \frac{1}{4} = 0
\][/tex]
4. Calculate [tex]\( c_{22} \)[/tex]:
[tex]\[
c_{22} = (1 \cdot \frac{1}{2}) + (-1 \cdot -\frac{1}{2}) = \frac{1}{2} + \frac{1}{2} = 1
\][/tex]
The resulting matrix [tex]\( AB \)[/tex] is:
[tex]\[
AB = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\][/tex]
This is the identity matrix [tex]\( I \)[/tex].
Since [tex]\( AB \)[/tex] is the identity matrix, the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are indeed inverses of one another.
So, the matrices [tex]\(\textbf{are}\)[/tex] inverses of one another.
