repmgr cluster crosscheck is similar to repmgr cluster matrix, but cross-checks connections between each combination of nodes. In "Example 3" in repmgr cluster matrix we have no information about the state of node3. However by running repmgr cluster crosscheck it's possible to get a better overview of the cluster situation:
$ repmgr -f /etc/repmgr.conf cluster crosscheck
Name | Id | 1 | 2 | 3
-------+----+----+----+----
node1 | 1 | * | * | x
node2 | 2 | * | * | *
node3 | 3 | * | * | *
What happened is that repmgr cluster crosscheck merged its own repmgr cluster matrix with the repmgr cluster matrix output from node2; the latter is able to connect to node3 and therefore determine the state of outbound connections from that node.