A bad computer (PCM, ECU) is the least likely of all possibilities.

Why did you replace the cam sensor? Were you getting a code? If so, what was it?

Also, you said "it tries to start". Maybe inspect the spark plugs. Make sure an oil leak(s) from the valve cover gasket isn't shorting out any connections at the COP (coils) to the spark plugs. 

Then hook up your scanner and look at Live Data. See if the temps which the ECT and IAT sensors are reporting look reasonable for a cold engine at the current ambient air temperature.

Then open the air filter box, make sure the filter is clean and since it's getting close to winter, make sure no critters took up residency in there. 

If all that checks out OK then it's time to see which of these 3 are missing:

1) Sufficient Spark (at the right time)

2) Air/Fuel (at the correct mixture)

3) Compression.

If you're sure you have good spark and that the engine has good compression then maybe starting with a possible fuel delivery issue is the place to begin.

While you have the air intake open, spray some starting fluid into the throttle body (throttle plate open) and see if the Matrix will start for a second or two.

If it does start, you can start tracking down the issue with the fuel delivery.

1) Low Fuel pressure (fuel pump module) More likely because "it tries to start".

2) Fuel injectors. Less likely because "it tries to start"

Since you don't have a schrader valve on the fuel rail of those Toyotas (to make it easy to test the fuel pressure), look online for how to test the fuel pressure on a Toyota without a Schrader valve.

Probably 100's of videos on how to do that and it isn't hard.

If you have good fuel pressure then it's time to test the fuel injector circuit.

Grab a test light and steal a straight pin from your wife's sewing basket.

You'll perform this test by "backpinning" the wires with that pin while the connector is still attached to each fuel injector.

Each injector has 2 wires going to its connector.

If you look, 1 of those wires will be the same color on all 4 injectors. That's your Power wire.

So, stab that wire at a fuel injector's connector with the sewing pin. Attach the alligator clip of the test light to the Negative battery terminal, turn the Key to ON. Touch the test light's probe to the sewing pin. The test light should light up.

If it does, the Power part of the circuit is fine. If it doesn't you likely have a fried fuse or bad relay on the Power side of the fuel injector circuit.

Next, test that the Ground switching signal from the ECU (PCM/computer) is reaching each fuel injector.

Move the test light's alligator clip to the Positive battery terminal.

Move the sewing pin and stab the other wire at the fuel injector's connector.

As you touch the test light's probe to the pin, have somebody crank the engine.

The test light should flicker ON/OFF.

If it does, the computer is providing the Ground switching signal and it's reaching that fuel injector (test at all 4 injectors).

If it doesn't, then either the computer isn't providing the Ground switching signal OR there's a wiring issue in the fuel injector harness.

That's easy to test too, but I'm tired of typing so let us know what you find out.