How to construct an injection of $PGL(n,k)$ in $PGL(n+1,k)$ if $PGL(n,k)$ injects in $PGL(n+1,k)$. I think it depends on the field k. For example, if we put $\varphi:GL(n,k)\longrightarrow GL(n+1,k)$ defined by : 

$$\varphi(g)=\left(
   \begin{array}{cc}
     g & 0 \cr
     0 & \chi(g) \cr
   \end{array}
 \right)$$

where $\chi$ is a caracter of $GL(n,k)$, that is a morphism of groups of $GL(n,k)$ to $k^{\times}$, then $\chi$ has the form $\chi=\phi\circ det$, where $\phi$ is an endomorphism of $k^{\times}$.

Then, $\varphi$ induces a morphisme of groups of $PGL(n,k)$ in $PGL(n+1,k)$ if and only if the morphism $\phi$ satisfies : For all $x\in k^{\times}$, $\phi(x)^{n}=x$. 

For $k=\mathbb{R}$ that is imposible.