sudo add-apt-repository "deb http://packages.mate-desktop.org/repo/ubuntu precise main"
printf "\nPackage: *\nPin: origin packages.mate-desktop.org\nPin-Priority: 700\n" | sudo tee -a /etc/apt/preferences
sudo apt-get update && sudo apt-get install mate-archive-keyring
sudo apt-get update && sudo apt-get upgrade
You might also want to install the mate-desktop-environment meta package, to bring in some new MATE components. It will install some additional backgrounds and themes, and give you the ability to send and receive files over bluetooth.
sudo apt-get install mate-desktop-environment