AB\widetilde{(AB)}=\det(AB)I \Rightarrow \tilde{A}AB\widetilde{(AB)}=\tilde{A}\det(AB) \Rightarrow \det(A)B\widetilde{(AB)}=\tilde{A}\det(A)\det(B) \Rightarrow \tilde{B}B\det(A)\widetilde{(AB)}=\tilde{B}\tilde{A}\det(A)\det(B) \Rightarrow \det(B)\det(A)\widetilde{(AB)}=\tilde{B}\tilde{A}\det(A)\det(B) \Rightarrow \det(A)\det(B)(\widetilde{(AB)}-\tilde{B}\tilde{A})=0