 Solution

Embed ${\mathrm{GL}}_n(F)$ by $A\mapsto (A,0)$. The multiplication and inverse formulas show that this is an injective group homomorphism, so its image $H$ is a subgroup. The translations form $N=\{(I,c):c\in F^n\}$, with $(I,c)(I,d)=(I,c+d)$ and inverse $(I,-c)$. Hence $N$ is isomorphic to the additive group of $F^n$.

For an arbitrary affine transformation,Conjugation therefore preserves $N$, proving that $N$ is normal in $G$.