Can the identity matrix be negative? I got the following question:
Find, if possible, the inverse of the matrix:
$\begin{bmatrix}3, -1\\2, -2\end{bmatrix} $  

and I did the following:
$\begin{bmatrix}3, -1\\2, -2\end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix}-2, 1\\-2, 3\end{bmatrix} = \frac{1}{4} \begin{bmatrix}-2, 1\\-2, 3\end{bmatrix}$  

Test: $( A * A^{-1} = I)$
$\begin{bmatrix}3, -1\\2, -2\end{bmatrix} *\begin{bmatrix}-0.5,0.25\\-0.5, 0.75\end{bmatrix} =  \begin{bmatrix}-1, 0\\0, -1\end{bmatrix}$  

Is this correct?
No, the identity matrix cannot be negative.  If your check yields $AA^{-1} = -I$ then something must have gone wrong.
La faute est dans le calcul du déterminant, on a : ad-bc=-4 et non 4

The error is in the calculation of the determinant. We have $ad-bc=-4$ and not $4$.