Bottom line: Yes, you can. Should you, is an entirely different question. I tackle this in a recent piece I posted over at Backblaze.
Time Machine is Apple’s backup software built into OS X, and it makes for simplified backup, version control and file recovery. If you upgrade your Mac or if you have to get it repaired, Time Machine makes it trivially easy to keep going where you left off.
SSDs, or Solid State Drives, are storage devices that use memory cells instead of “traditional” hard drives. They’re much faster than hard drives – faster to boot, faster to read, and faster to write. SSDs are a popular internal upgrade for older computers that still use conventional 2.5-inch SATA hard drives inside. You can also put together your own external SSD with USB 3 or even Thunderbolt 2 without spending a fortune, and you can buy a few pre-made ones too.
That got me thinking: Time Machine works using an external hard drive, and with SSD prices coming down, you may think that you’ll get faster backups with an SSD instead of a regular hard drive. You’d be right. But SSD performance is much more important for your actual boot drive than it is for backup media, which usually doesn’t have to happen quite so fast. SSDs are cheaper than they used to be, but regular hard drives still win the day when it comes to price per gigabyte.
Anyway, feel free to read on for more details, and let me know here or there if you have any questions.