Form the matrix of cofactors
Compute each minor by deleting the row and column of the entry
Apply the sign pattern ( (+, -, +; -, +, -; +, -, +) )
Transpose the cofactor matrix
The result is the adjoint
For a (2 times 2) matrix (begin{pmatrix} a & b \ c & d end{pmatrix}), the adjoint is (begin{pmatrix} d & -b \ -c & a end{pmatrix})