Let $F$ be a field, 
for $H \in M_{k \times k}(F)$, let $H^*$ be the adjugate matrix of $H$.

I am trying to prove the following two results:

1) If $H, G \in M_{n \times n}(F)$, then ${(HG)}^* = G^* H^*$.

2) If $H \in M_{n \times n}(F)$ and $G \in M_{m \times m}(F)$, then
\begin{equation*} \begin{pmatrix} H & 0 \\
0 & G
\end{pmatrix}^* = \begin{pmatrix}
\det(G) H^* & 0 \\
0 & \det(H) G^*
\end{pmatrix}.
\end{equation*}

My current idea, is that both of these results hold for invertible matrices (using that $H^* = \det(H) H^{-1}$, when $H \in GL_n(F)$). Then, using that the set of invertible matrices is Zariski dense $M_{n \times n}(F)$, deduce the result for all matrices.

This argument is very similar to the topological proof of the Cayley-Hamilton theorem.
To this end it would be useful to have a reference for a proof of the Cayley-Hamilton theorem using algebraic geometry.
Alternatively, does anyone know of a reference which proves the two results.