Are you SURE you want to take this on? It will not be easy!!! At least not taking your description of what you want to do at face value. There's probably a way to get your result, but not by the "obvious" route.
a) Do you have access to the 13 wires (6+7) by which the matrix is connected to it's host?
b) If you REALLY want to EXACTLY mimic "fingers on keys", you'll need to be able to connect 84 wires to the places on the matrix keyboard where it's SPST switch connects to the cirsuit board.
Those 84 wires could be taken to 84 opto-osolators which could be Arudino controlled, and act like "fingers on buttons".