For a 2×2 matrix (A=begin{bmatrix}a & b\ c & dend{bmatrix})
The adjoint is (operatorname{adj}(A)=begin{bmatrix}d & -b\ -c & aend{bmatrix})
Swap the diagonal elements
Change the signs of the off-diagonal elements
Example: if (A=begin{bmatrix}2 & 3\ 4 & 5end{bmatrix}), then (operatorname{adj}(A)=begin{bmatrix}5 & -3\ -4 & 2end{bmatrix})