Checking the correctness of the adjacency matrix for the given graph

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I found the adjacency matrix for this graph; it is shown next to it. Is it correct?

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Yes, the matrix is correct. As a side observation, the adjacency matrix being of the form $$ \begin{pmatrix} 0 & * \\ * & 0 \end{pmatrix} $$ tells us that we have a bipartite graph (or rather multigraph, since you allow multiple edges).