What resisters for inverting op-amp should I use?
I have a Clarion xc1410 4-channel amplifier with a 10k input resistance that is non-bridgeable. They are powering two 4 ohm speakers. The amp is capable up to 2 ohms. 


Make that "down to 2 ohms". 2 Ω is the minimum resistance allowed.


  I want to bridge the amp to create 2 channels producing twice the power. My understanding is if I invert the input signal on 2 channels I can bridge those channels with the non-inverted channels. 


You've got the basic principle of bridging correct.


  This is what my test circuit looks like:


Thanks for the link. That's a wiring diagram and they're generally despised on this site as they don't show the schema of the circuit. There's a nifty schematic editor tool on the editor toolbar. It takes a minute or two to see that you have connected pin 4 to the GND on the breadboard. That means that your op-amp output can never swing negative. You need a dual-voltage supply - e.g., +12 / 0 / -12 to allow the op-amp to handle alternating signals.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Inverting op-amp circuit with dual-rail supply.


  I'm not 100% sure what resistors to use as my input resistor and my feedback resistor to match this ideally with my 10k input resistance on the Clarion amp.


Notice the Inverted output on Figure 1 has no resistor between the op-amp and the Inverted output. This means that it has a very low output impedance - maybe less than 1 Ω and it will easily drive your amplifier input. So we choose a value for R1 that doesn't load the driving circuit (your pre-amplifier) and isn't so high that noise becomes a problem. 10k is fine. Since you want a gain of -1 R2 is 10k also.


  ... if I invert the input signal on 2 channels I can bridge those channels with the non-inverted channels. 


Here's where it gets tricky. With a 2 Ω output capability your amplifier may already be bridged internally. It seems unlikely, but you'd better check. If there are amplifier chips (on large heatsinks) then count them and try and figure out what's going on from the chip datasheets. i.e., see if they're single or dual amplifiers. If it's transistor then post a photo.







simulate this circuit

Figure 2. If your amplifier is not already bridged you should be able to identify the four GND terminals on the speaker connections. They should bell out (0 Ω to each other and chassis).

For testing connect up a 12 V car tail lamp or brake lamp in series with each speaker. If it glows at all with no music in then you've got a problem.
You should also check that there is no DC voltage across the speaker when volume is down. If all looks well wind the volume up gently. If that goes well then remove the lamps.



Update after car battery power mentioned:







simulate this circuit

Figure 3. Configuration for single-ended supply. OA2 provides a half-supply "virtual earth" for the op-amps. OA3 is added to avoid phase shifts on account of the DC blocking capacitors.

Add 100 nF decoupling capacitors to the power pins beside each op-amp.