Matrix inverse's existential crisis

Consider the matrix

\[A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\]

with \(P(a_{ij} = 0) = P(a_{ij} = 1) = \frac{1}{2}\) for all entries.

If the probability that \(A\) is invertible can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are positive coprime integers, find \(m+n\).

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