To solve the matrix equation
[tex]\[
\begin{pmatrix}
\frac{1}{2} & -\frac{1}{4} \\
2 & -\frac{3}{4}
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
2 & -4 \\
1 & 6
\end{pmatrix},
\][/tex]
we need to find the inverse of the matrix
[tex]\[
A = \begin{pmatrix}
\frac{1}{2} & -\frac{1}{4} \\
2 & -\frac{3}{4}
\end{pmatrix}.
\][/tex]
Let's denote the matrix on the right-hand side as [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix}
2 & -4 \\
1 & 6
\end{pmatrix}.
\][/tex]
### Step 1: Calculate the inverse of matrix [tex]\(A\)[/tex]
Given [tex]\( A = \begin{pmatrix}
\frac{1}{2} & -\frac{1}{4} \\
2 & -\frac{3}{4}
\end{pmatrix} \)[/tex],
the inverse of [tex]\( A \)[/tex] is
[tex]\[
A^{-1} = \begin{pmatrix}
-6 & 2 \\
-16 & 4
\end{pmatrix}.
\][/tex]
### Step 2: Find the matrix solution [tex]\(X\)[/tex]
To solve for [tex]\(X\)[/tex], we use the equation:
[tex]\[
A^{-1} A \begin{pmatrix}
x \\
y
\end{pmatrix}
=
A^{-1} B.
\][/tex]
Since [tex]\( A^{-1} A = I \)[/tex] (where [tex]\( I \)[/tex] is the identity matrix), the left-hand side simplifies to
[tex]\[
I \begin{pmatrix}
x \\
y
\end{pmatrix} = \begin{pmatrix}
x \\
y
\end{pmatrix}.
\][/tex]
Therefore,
[tex]\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
A^{-1} B.
\][/tex]
### Step 3: Calculate [tex]\( A^{-1} B \)[/tex]
Multiplying the inverse of [tex]\( A \)[/tex] by [tex]\( B \)[/tex]:
[tex]\[
A^{-1} B = \begin{pmatrix}
-6 & 2 \\
-16 & 4
\end{pmatrix}
\begin{pmatrix}
2 & -4 \\
1 & 6
\end{pmatrix}
= \begin{pmatrix}
-10 & 36 \\
-28 & 88
\end{pmatrix}.
\][/tex]
So the matrix solution [tex]\(X\)[/tex] is
[tex]\[
X = \begin{pmatrix}
x \\
y
\end{pmatrix} = \begin{pmatrix}
-10 & 36 \\
-28 & 88
\end{pmatrix}.
\][/tex]
Thus, we can summarize our solution to the matrix equation as
[tex]\[
\begin{pmatrix}
x \\
y
\end{pmatrix} = \begin{pmatrix}
-10 & 36 \\
-28 & 88
\end{pmatrix}.
\][/tex]
