To completely remove MATE:
Uninstall mate-desktop
To remove just mate-desktop package itself from Ubuntu 16.04 (Xenial Xerus) execute on terminal:
sudo apt-get remove mate-desktop
Uninstall mate-desktop and it's dependent packages
To remove the mate-desktop package and any other dependant package which are no longer needed from Ubuntu Xenial.
sudo apt-get remove --auto-remove mate-desktop
Purging mate-desktop
If you also want to delete configuration and/or data files of mate-desktop from Ubuntu Xenial then this will work:
sudo apt-get purge mate-desktop
To delete configuration and/or data files of mate-desktop and it's dependencies from Ubuntu Xenial then execute:
sudo apt-get purge --auto-remove mate-desktop
REF:LINK
sudo apt remove ubuntu-mate-*