Let $\mathbb K$ be a commutative ring with unity. Let $det:\mathbb K^{n\times n}\to \mathbb K$ be determinant function.
Prove that,
$det A$ is invertible $\iff$ $A$ has an inverse.
I proved this for $\mathbb K= a \, Field$. Because, in that case it reduces to $det(A)\neq 0$. But, how we will prove if $\mathbb K$ is s commutative ring with unity.
Work already done:
If $A$ has an inverse, then $AA^{-1}=I=A^{-1}A$ where $A^{-1}$ being the inverse of $A$. Hence, $$det(AA^{-1})=det(I)=det(A^{-1}A)$$ $$det(A)det(A^{-1})=1=det(A^{-1})det(A)$$ which proves that $det(A)$ is invertible.
Now, we have to prove that $det(A)$ is invertible $\implies A$ has an inverse.
If $A$ is invertible then $\exists \, M \in \mathcal{R}^{n \times n}$ such that $AM=MA=I_n$. Then determinant being a homomorphism gives $$\det(AM)=\det(A)\det(M)=\det(I)=1_{\mathcal{R}}.$$ Thus $\det(A)$ must be a unit in $\mathcal{R}$.
For the other way:
Assume $\det(A)$ is invertible. Since $$\det(A)I_n = A\operatorname{adj}(A) = \operatorname{adj}(A)A$$ therefore determinant being invertible implies the inverse exists, since the adjoint always exists.