LED Matrix Extras Lighting Up
I would have expected (C4, R1) to light as well.

You need multiplexing to solve this problem.  Only light one row (or one column) at a time.  Switch rapidly through the rows (or columns) to make it appear that they are all on at the same time.

If the display appears too dim, adjust the resistors down a bit.  You can get away with this if the individual LEDs are only on for a few milliseconds at a time.