Controlling a 19x19 input matrix
I suspect you are thinking of a keyboard scanning chip, but those chips aren't likely to be very happy with so many switches being actuated simultaneously.

These are your options as I see them:

1) If you're wedded to the Raspberry Pi ecosystem, use some configuration of the MCP23017.  That's likely to have the best Raspberry Pi support by a wide margin.  Apparently someone has already put two MCP23017 chips together into a HAT called the "IO Pi Plus" (you may be able to use 6 pins on the Pi header somewhere to get to the necessary 38 I/Os).  You could do the same thing with shift registers/serial-to-parallel/parallel-to-serial chips, but it's probably no cheaper or easier.

2) If you're not wedded to the Raspberry Pi, a Beaglebone Black has more than enough I/Os directly.

3) If you aren't afraid of microcontrollers, you could get a cheap Cortex M0 with enough I/Os that can be configured to do what you want with a fairly low amount of programming.  Use the SPI bus to communicate to the microcontroller, and you can make the microcontroller look like latching shift registers with a very tiny amount of code (probably less than 50 lines of code on the microcontroller).

4) For the "recast the problem" solution--don't use reed switches but instead use a Hall Effect Sensor with an I2C interface.  Something like the A1454 from Allegro MicroSystems can be programmed to be one of 127 addresses on the I2C bus, so you would only need 3 I2C buses (or one I2C bus properly multiplexed) to communicate to all of the sensors in the matrix.  This has the advantage that it is likely to be more robust to piece misalignment than a simple reed switch as you can actually measure the magnitude of the magnetic field.

Hope this helps.