Determine if these matrices are inverses by calculating [tex]\( AB \)[/tex]:

[tex]\[
A = \begin{bmatrix}
2 & 2 \\
1 & -1
\end{bmatrix}
\quad
B = \begin{bmatrix}
\frac{1}{4} & \frac{1}{2} \\
\frac{1}{4} & -\frac{1}{2}
\end{bmatrix}
\][/tex]

Calculate the elements of [tex]\( AB \)[/tex]:
[tex]\[
AB = \begin{bmatrix}
c_{11} & c_{12} \\
c_{21} & c_{22}
\end{bmatrix}
\][/tex]

[tex]\[ c_{11} = \][/tex]
[tex]\[ c_{12} = \][/tex]
[tex]\[ c_{21} = \][/tex]
[tex]\[ c_{22} = \][/tex]

So the matrices are \_\_\_\_ inverses of one another.



Answer :

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.