Result
R5505: Cofactor matrix for a 2-by-2 real square matrix shows that
\begin{equation}
\text{Cof} A
=
\begin{bmatrix}
d & - c \\
- b & a
\end{bmatrix}
\end{equation}
Therefore
\begin{equation}
\text{Adj} A
= (\text{Cof} A)^T
=
\begin{bmatrix}
d & - b \\
- c & a
\end{bmatrix}
\end{equation}
$\square$