Still, I don't understand the question. The matrix determines the graph, hence there is a way to tell whether it's connected. If the question is about how to do that, then here is the first thing that comes to my mind:

the double is connected iff (1) the original graph $G$ is connected, and (2) the homomorphism $H_1(G)\to\Bbb Z_2$ defined by the $2$-lift pattern is nontrivial.

Condition (2) can be restated as follows: there is a sequence of indices $i_1,i_2,\ldots,i_k,i_{k+1}=i_1$ such that $\prod_{j=1}^ka_{i_j,i_{j+1}}=-1$.

Still, I don't understand the question. The matrix determines the graph, hence there is a way to tell whether it's connected. If the question is about how to do that, then here is the first thing that comes to my mind:

the double is connected iff (1) the original graph $G$ is connected, and (2) the homomorphism $H_1(G)\to\Bbb Z_2$ defined by the $2$-lift pattern is nontrivial.

Condition (2) can be restated as follows: there is a $1$-cycle spanned by the original vertices $i_1,i_2,\ldots,i_k,i_{k+1}=i_1$ such that $\prod_{j=1}^k e(\{i_j\ i_{j+1}\})=-1$.

Still, I don't understand the question. The matrix determines the graph, hence there is a way to tell whether it's connected. If the question is about how to do that, then here is the first thing that comes to my mind:

the double is connected iff (1) the original graph $G$ is connected, and (2) the homomorphism $H_1(G)\to\Bbb Z_2$ defined by the matrix is nontrivial.

Condition (2) can be restated as follows: there is a sequence of indices $i_1,i_2,\ldots,i_k,i_{k+1}=i_1$ such that $\prod_{j=1}^ka_{i_j,i_{j+1}}=-1$.

Still, I don't understand the question. The matrix determines the graph, hence there is a way to tell whether it's connected. If the question is about how to do that, then here is the first thing that comes to my mind:

the double is connected iff (1) the original graph $G$ is connected, and (2) the homomorphism $H_1(G)\to\Bbb Z_2$ defined by the $2$-lift pattern is nontrivial.

Condition (2) can be restated as follows: there is a sequence of indices $i_1,i_2,\ldots,i_k,i_{k+1}=i_1$ such that $\prod_{j=1}^ka_{i_j,i_{j+1}}=-1$.