Here is how the metering works when using the Nikon Built-In Flash.
When you select camera Matrix metering the built-in flash automatically switches to TTL-BL mode.
The question I will try to answer is: Why does it do this?
Well, when you use Matrix metering, the camera first meters the brightness of the ambient light of the entire frame fairly evenly from edge to edge. Then, when a subject is placed in a small portion of the center of the frame, you can see that it does not cause a significant change to the matrix measurement made by the camera. In other words, in matrix mode, the measurement that the camera makes is a very good approximation of the background ambient light BEHIND the subject.
This approximation concept is the fundamental basis for the TTL-BL mode!
Next, the built-in flash fires its preflashes, and it measures the strength of the reflection only in the center portion of the frame, because it is trying to measure the reflection from only the subject. In other words, the flash metering system always assumes there is a subject centered in the frame when it fires its preflashes.
Then, the flash computer compares the brightness of the subject (based on the reflected preflash from the center of the frame) to the brightness of the background (based on the matrix measurement of the entire frame from the camera), and it computes a flash power that will make the subject brightness equal to the overall scene brightness. Then the shutter is opened and the main flash is fired and the picture is taken.
Now, for the pop-up flash Nikon wanted to simplify things, so they forced the flash to always operate in TTL-BL whenever the camera is in Matrix mode. This relieves a novice photographer from ever having to make the fairly complex decision of when to use TTL mode and matrix together, which is allowed when using an external flash in the hotshoe.
Also, it's important to understand that the choice of the camera metering mode has absolutely no effect on the way the flash meters the preflash reflection. It always looks at the center of the frame regardless of what camera metering mode is chosen.
Now, if you switch the camera to spot metering, the camera now only measures the brightness of the subject due to ambient light; ie there is no background metering data, so obviously there is no way for the flash to work in TTL-BL mode when the background information is missing.
So the flash automatically switches to TTL mode, and no additional metering information is sent from the camera to the flash.
In TTL mode the flash uses only its own reflected preflash information, to set a flash power to make the brightness of the center of the frame (the subject) be a 'standard' brightness. This 'standard' brightness is adjusted in the factory to properly expose a subject of normal reflectivity placed in the exact center of the frame.
The size of the flash metered area is exactly the same whether in TTL or in TTL-BL modes. The size of the camera metered area changes depending on which camera metering mode is chosen.
Now, the pop-up flash can also work in TTL-BL when the camera is in Center Weighted mode, but this will only work correctly under certain circumstances. One is when the subject occupies only a very very small portion of the center of the frame, so that the camera metering information that is sent to the flash contains at least a little background information. Then the flash computer runs the TTL-BL calculations assuming the data from the camera is background information. It doesn't work very well most of the time, because the subject usually occupies too much space in the center of the frame and affects the background information substantially.
In summary, you may be able to see that there are a few situations that won't meter very well when using the built-in flash coupled with matrix metering, because you are forced to use TTL-BL. In fact, indoors the built-in flash will often work much better if you select Spot metering which forces the flash to function in TTL mode. Matrix mode will usually work better outdoors than indoors, because the background outdoors is generally brighter than the subject, and with the flash in TTL-BL the flash will add brightness to the subject to make it equal the background. Matrix and TTL-BL will not work well if the subject is brighter than or equal to the the background. That's when you need to switch to Spot and TTL.
This last situation highlights one of the big advantages of an an external flash, where you can leave the camera in Matrix and switch the flash manually to TTL.