Matrix expression of the scalar product between two vectors

Question

My question is in regarding the adjugate matrix in the expression my teacher gave me of the scalar product: $$ \vec{x}\cdot \vec{y}=x^+ G y$$ being $x^+=\overline{x}^T$ the traspose of the conjugate, in the complex plane. I know that in the real plane, $x^+=x^T$ so the expression becomes $x^T Gy$ as everyone is familiar with. But the thing is that if this is true, why when finding the inverse of a matrix $A$ (let's assume now all the elements are real of this matrix) with cofactors the expression is: $$A^{-1}=\frac{1}{\text{det} A} (A^+)^T$$ and until now, in that expression the adjugate was used finding the cofactors as in with each element of it being $A_{ij}=(-1)^{i+j} \alpha_{ij}$ and then transposing it, while due to the logic before, since all terms are real it should just be $x^+=x^T$? . Have I been conflicting two different terms or how are both things the same?