 Solution

Each diagonal entry of $A$ is $-6$. Two distinct grid unknowns have matrix entry $1$ precisely when their nodes are nearest neighbours in one coordinate direction, and otherwise have entry $0$. Nearest-neighbour adjacency is symmetric: node $i$ is adjacent to node $j$ exactly when node $j$ is adjacent to node $i$. Hence $A_{ij}=A_{ji}$ for every pair, independently of the chosen ordering, and