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From page 17 of Algebraic properties of Manin matrices, I deduce thatpage 8:$(\det_q M)(\det_q M^{-1})=\det_q(MM^{-1})=1$, so the inverse of the determinant is the determinant of the inverse$(\det_q M)^{-1}=\det_q(M^{-1})$.
From page 17 of Algebraic properties of Manin matrices, I deduce that$(\det_q M)(\det_q M^{-1})=\det_q(MM^{-1})=1$, so the inverse of the determinant is the determinant of the inverse.
From page 17 of Algebraic properties of Manin matrices, I deduce that $(\det_q M)(\det_q M^{-1})=\det_q(MM^{-1})=1$, so the inverse of the determinant is the determinant of the inverse.
