For square $A$, let $A_{\widehat i,\widehat j}$ be the submatrix obtained by deleting row $i$ and column $j$. The corresponding minor is $\det A_{\widehat i,\widehat j}$ and cofactor is $C_{ij}=(-1)^{i+j}\det A_{\widehat i,\widehat j}$. A submatrix and its determinant are different objects. The adjugate is the transpose of the cofactor matrix.
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