Let $k$ be an algebraically closed field, then it is well-known that any affine algebraic group $G$ over $k$ can be viewed as a closed subgroup of $\mathrm{GL}_n$ for some $n$.
In the special case that $G=\mathrm{PGL}_2$, how to do this explicitly on the $k$-rational points, i.e. if \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} is a representative of some element $g$ in $G$, how to write $g$ as a matrix in some $\mathrm{GL}_n$, preserving the group operation?