Thank you for your question. You are able to add an additional computer to iTunes Match. Your Apple ID can have up to 10 devices and computers (combined) associated with it.
I recommend following the steps under the section titled "To add a computer to iTunes Match" in the article below to set up iTunes Match on your alternate computer:
When you add a computer to iTunes Match using the steps in the article listed above, your content from that additional computer will get added to your iTunes Match account:
Once iTunes has completed matching and uploading your iTunes library, your music in iCloud will be integrated into your local music library. A download button will appear to the right of any content that you have in iCloud that doesn't already exist on your computer.
If there are issues with this process, you may find more information by using another one of Apple's support resources - https://getsupport.apple.com/GetproductgroupList.action
All the best,