I'm reading "Differential Geometry: Curves and Surfaces" of Manfredo Carmo, and this part in the book confuses me(page 166):
Suppose that $N: S \rightarrow S^2$ is the Gauss map of regular surface $S$ into unit sphere. We know that for each $p \in S$, $dN_p$ is a self-adjoint linear mapping. Suppose that $\{w_1, w_2\}$ is a basis in $T_p(S)$, then we have: $$dN_p(w_1) \times dN_p(w_2) = \det(dN_p)(w_1 \times w_2) = Kw_2 \times w_2$$
So the part confuses me is the equation $$dN_p(w_1) \times dN_p(w_2) = \det(dN_p)(w_1 \times w_2)$$
I did search around, and found this equation for cross product: $$Ma \times Mb = \det(M)(M^T)^{-1} (a \times b)$$
So, to make the mentioned equation right, we need to have $(M^T)^{-1} = I$, which I don't think it's true in general. Can anyone help me explain this? I really appreciate.
The condition you've arrived at is exactly the "self-adjoint linear mapping" condition in the passage.