Another technician came out and he tested this and that and told me that the motherboard in the computer was broken, so I would have to buy a new computer. This was on Tuesday morning, and I had a spare 90 minutes between appointments so I got in the car and went down to the electrical store at Broadway Shopping Centre and bought a new HP. The technicians sent out a new guy to install it and bring across the data files from the old one, but in the process of doing that work he found that the screen was still not working. Therefore the problem of the blinking screen turned out to be either with the video cable or with the monitor.
I got in the car and the technician got on his motorbike this morning and we went down to an electronics store near Ultimo TAFE and I bought a new monitor. We brought it back and installed it and it worked fine - I now had a new PC and a new monitor - and it was also bigger than the previous one, so I have abundant screen room now. It was also cheaper than the previous one - which, admittedly, had been bought in 2009 - so I was marginally happy. The new technician - who had done work on two days for me - called his head office and they discussed my case and decided to waive the fee for the second technician, so I will only pay for the first technician.
This is a good outcome anyway because even though the computer was not broken, most computers only last for about 4 or 5 years, so it would have been time to get a new one soon anyway. I gave the second technician the old computer and monitor in case he could get the monitor to work. He took them home to his place in his car.