Matrix - Visual Example

Nodes: $1$, $2$, $3$. Edges: ($1$-$2$) and ($2$-$3$). Here is what the matrix looks like:

  1 2 3
1 [ 0 1 0 ]
2 [ 1 0 1 ]
3 [ 0 1 0 ]

Notice the matrix is symmetric. Position M[1][2] equals position M[2][1] because if node $1$ connects to node $2$, then node $2$ connects to node $1$. This symmetry only holds for undirected graphs. For directed graphs, the matrix is not symmetric.

The diagonal is all zeros because no node connects to itself. If self-loops existed, you would see ones on the diagonal. You can read any cell to immediately answer "Is there an edge between $i$ and $j$?" That lookup takes $O(1)$ time, which is the main advantage of this representation.