You can add any type of file to a Git repository. However, if you are using command-line Git (meaning, you have installed Git on your own to use it at the command line instead of using the integration in the Current Folder browser), you have to make sure to configure Git so that .mat files and other binary files are recognized as binary files. Otherwise, Git might try to change the end-of-file character which will corrupt the file.
This is explained in details in the documentation:
