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Black dandruff is usually what is called flea dirt. It may be you brought a flea home on your clothes that found her a good place to be or if you have pets that do go out they could have brought some home too.
To check to see if that is what this is you can comb some out onto a white paper towel or sheet of paper and apply a few drops of water and see if the water turns reddish in a few minutes.
Fleas feed on blood so their waste or dirt will show evidence of that when moistened.
If that is the problem you can use Frontline on your cat and spray the floors of your home using an insect growth regulator such as Precor - you can see one product with this here
If it is fleas you may need to also check your cat for tapeworms which leave egg segments that look like grains of rice in stool or on beds or around the anal fur.
Another possibility is that your cat has a sebaceous gland issue which you can read about here
Hope this helps you!