Can it work?
ROW pins pull high when active and high impedance when passive, the ULN2803 need input current to turn on, this looks acceptable.
COM pins pull low when active and is high impedance when passive, there is a non-inverting driver with 5V CMOS input and output. A floating input on a CMOS will often be staying in the previous state, i.e. low. The transistor input will be either 0V or 5V, but the transistor is supplied from 12V, this means it would be on all the time if it was a P-Channel transistor, but symbol shows N-Channel transistor wired to always pass current.
This is not going to work.
The problem is the two different supply voltages. A real mos driver chip could do it or a transistor more, but a transistor includes an inversion that you do not need, this has to be fixed with a transistor more or a inverter and you need a pullup on the COM outputs.