Electronic – the purpose of this diode

In your actual circuit, have you connected the wires and the LED leads at the junctions indicated by the black circles, or have you connected them wherever the lines overlap in your schematic diagram?

The correct reading of the schematic is to only have connections where indicated by the junction indicators (black filled circles). If you do so, then the third scenario in your diagram does not exist - the wires going horizontally and vertically do not actually touch at all.

One way to visualize this: The horizontal wires are all on the plane of your work table, the vertical wires are all in a plane floating an inch above the table, and the LEDs are the only connections between these two sets, and the only connections are the black circles.

You are correct, electricity will follow a path of least resistance to the greatest extent. In the schematic, there is no path of least resistance, i.e. short-circuit, between row and column wires, since no junctions are indicated. If you have interconnected the row and column wires, disconnect them now. Then, as you will see, the only path that allows electricity flow is through the respective diode / LED.

The schematics looks fine, and MCP6002 should be able to drive LED without any problem (it has I_SC of 23mA and V_OL of 25mV). Most likely the circuit is mis-assembled. You need to troubleshoot it:

1. Check opamp output. It should be <2 volts, ideally <0.1 volts.
2. Measure your power supply to make sure it is really giving you 5 volts.
3. Check voltage on the connection between resistor and LED. It should be ~1.5-2 volts. More than that means LED is broken or backwards or opamp is bad. Less than that means resistor is of incorrect value, or power supply is bad.
4. Check voltage across LED. It should be ~1.5-2 volts. If it is close to 5 volts, the LED is broken or installed backwards. If you have problems here, but steps 1-3 check out fine, you have power bus oscillations in your system.