Answer
$$A =\left[ \begin {array}{cc} \frac{1}{4}&-1\\ 0&\frac{1}{2}\end {array}
\right]
.$$
Work Step by Step
Since
$$((2A)^{-1})^{-1}=2A,$$
then by calculating the inverse of the matrix $ \left[ \begin {array}{cc}2&4\\0&1 \end {array} \right]
$ we have
$$2A=\left[ \begin {array}{cc} \frac{1}{2}&-2\\ 0&1\end {array}
\right]
.$$
Hence, $A$ is given by
$$A=\frac{1}{2}\left[ \begin {array}{cc} \frac{1}{2}&-2\\ 0&1\end {array}
\right]=\left[ \begin {array}{cc} \frac{1}{4}&-1\\ 0&\frac{1}{2}\end {array}
\right]
.$$
