Possible bug while calculating determinant?

Although (like @Sukera) I cannot replicate this using a recent Julia version, note that floating point arithmetic has some special properties you should be aware of.

If you are following along the calculations in a linear algebra text, you may want to convert to a type that is closed under the operations required for the determinant, eg

julia> det(Rational.(a))
0//1

(Incidentally, I would recommend textbooks that place less emphasis on the determinant).