Does the test menu show up OK?
Most probably the CPS-A01 ASIC on the A board is faulty. However, before throwing the A board in the spare parts bin, there are couple of other things you may check;
1) Check the back side of the A board. Specifically the area where the RAM chips are located (If it is a dash, these chips are the slim chips with single line of legs). The legs poking out of their solder points left too long at the factory and most often they may be touching each other. Examine carefully.
2) Check the A board to B board connectors. Check if there are missing/broken or bent pins.
3) If all checks fail, reflow the solder of the legs of the CPS-A01 chip carefully. There may be cracked solder points. After reflowing, check with magnifier that there are no solder bridges. Use lots of no-clean flux while reflowing to avoid solder bridges.
4) If THAT does not fix it, you can replace the CPS-A01 ASIC with a working one from a CPS2 A board. The main reason to kill a CPS2 A board is that in general CPS2 A-B boards can be easily swapped as if a game cartridge where swapping CPS1 A-B boards is not that easy. I mean, to me, CPS1 system is a complete PCB (A-B-C). I don't like to keep them seperated, it just doesn't sound right
But I do have a small collection of CPS2 systems that having only a few A boards does not bother me much. Again, this is me...
Good luck...