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The phrase *degenerate circuit* does not have a strict meaning out of a specific context. It means that in a certain case, something about the circuit differs in a significant way from the equations that apply to the more general case. Usually this means that, though it may still be possible to use the equations you have developed, they may no longer be the best way to analyze the circuit.

This meaning of degenerate is discussed in more detail at: Degeneracy (mathematics) on Wikipedia.

There is another related meaning of degenerate that you may encounter, which is a special case of the previous definition -- a degenerate matrix is one which is not invertible (Invertible matrix, Wikipedia). This can come up when solving a system of linear equations (KCL or KVL) if you mistakenly use the same equation twice, for example.