If your airbag light is on or flashing (beyond the normal 'on' period when you first start the car) you have come to the right post.
If you have a 2003-2004 Matrix, first check [USA] to see if you qualify for a free recall repair. Canadian VIN check here. If not, I recommend you check your VIN for recalls anyway. Many serious recalls have been issued for the Matrix, some even involving the airbags, just do not expect the other recall(s) to fix your airbag light.
If the repair will be on your dime it is time to roll up those sleeves and read the code. Here is a video showing how to read the code by connecting pins 4 and 13 in your OBDII terminal, turn the key, and count the flashes your airbag light makes.
The service manual has a written explanation of how to read the code, and then goes on to include a trouble shooting guide for the 2004 Corolla, which should apply to the Matrix from 2003-2008. If the previous link listing the codes is no longer available click here for a list of ALL Toyota codes.
Service manuals are great if you are the first one with the problem, but what are the 'real world' experiences with Matrix airbag codes?...
Many report that reflowing the solder on the contacts in the Matrix- Corolla clock repairs not only the dimming clock display problem but also repairs some airbag light issues. If your code leads you to "Airbag sensor assy center" (and possibly other fault codes) this might be the fix for you. The clock dimming issue seems to be caused by poor connections on the largest chips on the board. If you try this intending to fix the airbag light, try to reflow ALL of the connections on the board. Unless you have bionic vision, you may need to up your game with some visual aids such as this, or a pair of these. This is not a soldering project for a novice, and a mistake could kill you.
Other common repairs require replacing the "clock spring." If your code returns the possibility of needing a "Spiral cable sub–assy" then this fix might be for you. This is a device behind the steering wheel (click for a how-to replace video) which connects your horn, airbag, cruise and other buttons through to the steering column even though the wheel has to be able to turn. Your part number will differ depending on what features are embedded in your steering wheel so be careful to order the correct part.
Sometimes a replacement airbag computer is required. The airbag computer hides under the center console. I could not find a video or pictures of this so if you do this please document it for all and share. This module has a big brother memory which fills up and flags the airbag light, so if you get this computer from a parts yard, either get one from a car which was not wrecked, or be ready to figure out how to clear the memory. I did not find these instructions to be clear but they are the only instructions I could locate. When seeking a module on internet postings make sure you are obtaining the correct module and not a module reset "service."
I did not take much time in selecting the product links used on this page. Shop around for price. Some clock springs are available direct from China for $11.
Do any of this at your own risk. The airbag system contains explosive charges. Any mistake could injure or kill you or an occupant during an improper repair or after an improper fix.
That is all I know. It is not everything there is to know but it assembles most of what is known across the internet and puts it in one place. I hope this helps.