Sorry for the delayed reply, just got home from the night shift
What value was the resistor?
It's a 1k, the picture is a little fuzzy but I just grabbed 1k since I was only testing the LEDs for a brief period.
It strikes me looking at the photograph that you have no way to access the top row of connectors with your probes. Is this exactly how you wired it? Moving it down a row would be better.
The picture is a little off angle, but as it sits there are three rows of access to the bottom pins (C D and E of the lower bread board), and two rows of access to the top pins (F and G of the upper bread board). I could move it down a row but it would just mean two rows of access to the bottom pins, and three rows of access to the top pins.
With just 1 pin high and 1 pin low, there is no way to have multiple LEDs turn on.
That's what I was thinking too, but I was getting several lit LEDs (of varying brightness) with just one pin high and one pin low. So you think that does mean there are some weak connections within the matrix itself? Nothing to do but get a new matrix?
Heading off to dreamland, will check back in later in the day.
Thanks for all the help everyone has already offered! 