Using iTunes Match even though I never subscribed?

So I have a bit of a problem.

On the computer I'm using right now, my laptop, I have access to iTunes Match, a service I never subscribed for. At least, I think I have. I can see all of my purchases even though I haven't downloaded most.

However, on my desktop computer, I cannot see anything in the cloud unless I go to the iTunes Store and look at my Purchased songs.

Is there any way for me to see my purchases on my desktop computer's library or will I eventually have to subscribe to iTunes Match.