$\mathbb{P}GL_{n}$ is homotopy equivalent to the compact group $\mathbb{P}U_{n}$, whose $\mathbb{Z}/p$-cohomology are known:
What is the integral cohomology of $\mathrm{PGL}_n(\mathbb{C})$ as a space?
In particular, $H^2(PU(n);\mathbb{Z})=\mathbb{Z}/p$.
Consider a linear vector bundle $\eta\to\mathbb{P}GL_{n}=\mathbb{P}GL_{n}(\mathbb{C})$, whose fiber over the point $[A]\in\mathbb{P}GL_{n}$, $A=(a_{j,k})\in GL_{n}(\mathbb{C})$, is the complex line consisting of matrices $\lambda\cdot A=(\lambda\cdot a_{j,k})$, $\lambda\in\mathbb{C}$. What is the Chern class $c_{1}(\eta)$ of it? Is it non-trivial?
Similar questions for the inverse $\eta^{-1}$ (with fiber $\lambda\cdot A^{-1}$ over $[A]$), transposed $\eta^t$ (with fiber $\lambda\cdot A^t$ over $[A]$).
As is explained in the link, there is a principal $U(n)$-bundle $PU(n)\to\mathbb{C} P^{\infty}$. Is there any explicit description of the corresponding linear vector bundle over $PU(n)$?