Is the formula $A^{-1}=\frac{1}{\det (A) }\text{Adj}(A)$ really true for 2 x 2 matrices?

Question

Given a 2 x 2 matrix $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, we have the inverse formula $A^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.

But isn't $\text{Adj}(A)$ equal to $\begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$? So don't we have $\frac{1}{\det (A)}\text{Adj}(A)=\frac{1}{ad-bc}\begin{bmatrix} d & -c \\ -b & a \end{bmatrix} \ne \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$?

Answer 1

The formula is true for all square invertible matrices. For adjoint A you have to take transpose of 'cofactor matrix of A'. You have not taken transpose.