could you explain why the matrix is wired incorrectly?
No I said the ULN2003 is wired up all wrong not the matrix.
I also said that:-
The outputs should go to the matrix not the inputs, the ground should go to pin 8 not pin 10.
That is the outputs of the ULN2003 should be wired up to the LED matrix not the inputs of the ULN2003 to the matrix.
You should connect pin 8 of the ULN2003 to ground, you have connected it to pin 10.
The ULN2003 looks a lot better.
Now the matrix is not wired like the data sheet is it.
The data sheet shows pins 9, 14, 12 & 8, 1, 7, 2 all being anodes. These anodes should be connected to the arduino outputs.
The data sheet shows pins 13, 3, 4 & 11, 10, 6 as all being cathodes these should be connected to your ULN2003.
Not much, that IC is a darlington transistor array so it already contains the base resistor you would need if you used a normal transistor. It is smaller so there is less wiring up.
The LED wiring is off still.
Note that the pins are shown as going across the short length of the body. So you may want to plug the chip in across the middle divider.
I have labelled them A1-7 for the anodes,
and C1-5 for the cathodes.
A1 is pin 9
A2 pin 14
A3 pin 8
A4 pin 4 (also available on pin 12)
A5 pin 1
A6 pin 7
A7 pin 2
C1 pin 13
C2 pin 3
C3 pin 4 (also available on pin 11)
C4 pin 10
C5 pin 6
If you look on the bottom of the part you may see a 1 next to one of the corner pins.
Plug it in and drive the output pin for C1 high (display pin 13) and for R1 low (display pin 9) - should see the corner LED turn on. If not, rotate the display 180 degrees around and try again.
You think you have but you have missidintified something.
What is needed is a schematic of what you are trying to acheave, not the sort of thing you have been posting so far and then a clear photograph of what you have done, along with the code you actually run not something like it.
Otherwise how else can we identify what you have wrong.