AdjointMatrix[mat_?MatrixQ]:=Block[{temp},
temp=Map[Reverse,Minors[mat],{0,1}];
Transpose@temp*Table[(-1)^(i+j),{i,Length@mat},{j,Length@mat}]
];
(* Example *)
A={{1,2,3},{4,5,6},{7,8,9}};
B={{1,2,3,0,1},{2,1,-2,3,1},{5,4,3,1,3},{0,0,1,0,2},{1,5,-1,2,1}};
AdjointMatrix[A] // MatrixForm
AdjointMatrix[B] // MatrixForm
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