Is there a nilpotent and idempotent pair such that their product is the identity matrix?

Question

Trying to gain a better understanding of linear algebra. Is there such a nilpotent matrix ($A^2=0$) and idempotent matrix ($B^2=B$) such that their product is equal to the identity? $$AB=I$$ and/or $$BA=I$$ Similarly, can nilpotents and idempotents be each other's inverses? If not matrices, are there any other mathematical objects that may have one or all of these properties?

Answer 1

If $AB = I$ and $A^2 = 0,$ then multiplying $A$ on the left of $AB = I$ would give $$0 = 0B = AAB = AI = A$$ and then $$I = AB = 0B = 0$$ so this would require the identity object to equal the zero object.