I would like to show that the total space $X$ of the rank $2$ vector bundle $E=\mathscr O_{\mathbb P^1}(-1)\oplus\mathscr O_{\mathbb P^1}(-1)$ on $\mathbb P^1$ has trivial canonical bundle $\omega_X$.
So we have the structure morphism $p:X=\textbf{Spec }(\textrm{Sym }E)\to \mathbb P^1$, and the exact sequence $$0\to p^\ast\Omega_{\mathbb P^1}\to \Omega_{X}\to \Omega_{X/\mathbb P^1}\to 0,$$ which may be rewritten as $$0\to \mathscr O_X(-2)\to \Omega_X\to \Delta^\ast(I/I^2)\to 0,$$ where $I\subset \mathscr O_{X\times_{\mathbb P^1}X}$ is the kernel of the canonical map $\mathscr O_{X\times_{\mathbb P^1}X}\to \Delta_\ast\mathscr O_X$ induced by the diagonal morphism $\Delta:X\to X\times_{\mathbb P^1}X$. We have $$\omega_X=\wedge^3\Omega_X=\mathscr O_X(-2)\otimes \wedge^2\Delta^\ast(I/I^2)=\mathscr O_X(-2)\otimes\Delta^\ast\bigl(\wedge^2(I/I^2) \bigr),$$ but I am not able to show that $\wedge^2(I/I^2)\cong \mathscr O_{X\times_{\mathbb P^1}X}(2)$.
Thank you for your help!