counterexample related to vector bundles Let E be a vector bundle of rank 2 over a variety X. Is there a counterexample so that  $E$ is not isomorphic to $E^*\otimes det$ $E$?
 A: As Jason Starr wrote in his comment, every rank $2$ vector bundle $E$ is isomorphic to $E^* \otimes \det E$. Let me give a proof of this fact. 
Assume that $E$ is defined by the transition functions $$g _{\alpha\beta} \colon U_{\alpha} \cap U_{\beta} \longrightarrow \mathrm{GL}(2, k), \quad g_{\alpha \beta} = \begin{pmatrix} g_{\alpha \beta}^{11} & g_{\alpha \beta}^{12}  \\ g_{\alpha \beta}^{21} &  g_{\alpha \beta}^{22} \end{pmatrix}.$$
Then $E^* \otimes \det E$ is defined by the transition functions $$h_{\alpha \beta} = {}^tg_{\alpha \beta}^{-1} \cdot \det g_{\alpha \beta}= \begin{pmatrix} \,\,\,\,\,g_{\alpha \beta}^{22} & -g_{\alpha \beta}^{21}  \\ -g_{\alpha \beta}^{12} &  \,\,\,g_{\alpha \beta}^{11} \end{pmatrix}.$$
Now consider the family of constant functions $$f_{\alpha} \colon U_{\alpha} \longrightarrow \mathrm{GL}(2, k), \quad f_{\alpha} = \begin{pmatrix} \,\,\,\,0 & 1  \\ -1  &  0 \end{pmatrix}.$$
It is immediate to check that $$h_{\alpha \beta} = f_{\alpha} g_{\alpha \beta} f_{\beta}^{-1},$$
so $\{g_{\alpha \beta}\}$ and $\{h_{\alpha \beta} \}$ are equivalent sets of transition functions, and this in turn means that they define isomorphic vector bundles.
