The command that will allow the administrator to determine the first command that is executed inside the container right after it starts is docker inspect <container_id>. This command will display detailed information about the container, including its configuration, state, network settings, mounts, and logs. One of the configuration fields is "Entrypoint", which shows the command that is executed when the container is run. The entrypoint can be specified in the Dockerfile or overridden at runtime using the --entrypoint option.