9X9 RGB Led matrix, design suggestion
Make the matrix from WS2812B instead, then just serially data chain the data from one to the next for all of them.  Sparkfun carries thru hole 5mm parts in 2 different shapes. Then you don't need complex multiplexing either, just send out 240 bytes of data when you need an update using the FastLed.h library.

Or you can get strings of SMD WS2812Bs and lay out 9 strips of 9 LEDs, makes the wiring even simpler - connect the Gnd, Power all along one end, and a single data line from the end of one strip to the start of the next.