The adjugate is $\operatorname{adj}(A)=C^T$, the transpose of the cofactor matrix. Cofactor identities imply $A\operatorname{adj}(A)=\operatorname{adj}(A)A=(\det A)I$. Thus, if $\det A\ne0$, $A^{-1}=\operatorname{adj}(A)/\det A$. The adjugate is not the conjugate transpose.
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