Hi,
Sounds like a busted rom chip to me, you can pick up a replacement rom on ebay, but go for the one to add a hard drive, the 2.04 version rom will not work on an A600, cost should be at most £10. Other's suggestions regarding the motherboard seem unfounded as there is power, as indicated by the power and drive leds', check also floppy cable is properly attached at both ends. If you retry the original rom chip make sure that no legs are bent or broken and everything seats correctly into the socket before powering up the motherboard.
:-o :-o Flash