To find matrix [tex]\( X \)[/tex], given [tex]\( \frac{1}{2} X = A \)[/tex], we need to isolate [tex]\( X \)[/tex]. We can do this by multiplying both sides of the equation by 2:
[tex]\[
2 \left( \frac{1}{2} X \right) = 2A
\][/tex]
This simplifies to:
[tex]\[
X = 2A
\][/tex]
Given matrix [tex]\( A \)[/tex]:
[tex]\[
A = \left[\begin{array}{cc} 6 & -12 \\ 4 & 8 \end{array}\right]
\][/tex]
We multiply each element of matrix [tex]\( A \)[/tex] by 2 to find matrix [tex]\( X \)[/tex]:
[tex]\[
X = 2 \times \left[\begin{array}{cc} 6 & -12 \\ 4 & 8 \end{array}\right] = \left[\begin{array}{cc} 2 \times 6 & 2 \times -12 \\ 2 \times 4 & 2 \times 8 \end{array}\right]
\][/tex]
This results in:
[tex]\[
X = \left[\begin{array}{cc} 12 & -24 \\ 8 & 16 \end{array}\right]
\][/tex]
We now compare this result with the provided options:
1. [tex]\(\left[\begin{array}{cc}3 & -6 \\ 2 & 4\end{array}\right]\)[/tex]
2. [tex]\(\left[\begin{array}{cc}5.5 & -12.5 \\ 3.5 & 7.5\end{array}\right]\)[/tex]
3. [tex]\(\left[\begin{array}{cc}6.5 & -11.5 \\ 4.5 & 8.5\end{array}\right]\)[/tex]
4. [tex]\(\left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right]\)[/tex]
The correct matrix [tex]\( X \)[/tex] matches option 4:
[tex]\[
\left[\begin{array}{cc} 12 & -24 \\ 8 & 16 \end{array}\right]
\][/tex]
Thus, the correct answer is option 4.
