Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field. a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$. b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral. c) Deduce that the result of a) still holds for every commutative ring $A$. d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have $$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$
Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field.\\
a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$.\\
b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral.\\
c) Deduce that the result of a) still holds for every commutative ring $A$.\\
d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have
$$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$