Answer
$A=(-1/24)*\left[\begin{array}{cc}
8&-4\\
-6&2
\end{array} \right]$
Work Step by Step
Given $(2A)^{-1}=\left[\begin{array}{cc}
1&2\\
3&4
\end{array} \right]$
thus $ (1/2)A^{-1}=\left[\begin{array}{cc}
1&2\\
3&4
\end{array} \right]$
$A^{-1}=\left[\begin{array}{cc}
2&4\\
6&8
\end{array} \right]$
Since $A=(A^{-1})^{-1}=(1/|A^{-1}|)*\left[\begin{array}{cc}
8&-4\\
-6&2
\end{array} \right]=(-1/24)*\left[\begin{array}{cc}
8&-4\\
-6&2
\end{array} \right]$
