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Given a commutative ring $A$ with identity, the determinant function $\det_n: \mathcal{M}_n(A)\to A$ can be characterized as the unique $n$-linear, alternating function such that $\det(\textbf{I}_n)=1_A$. This characterization, however, involves properties relating to specific rows or columns of a matrix. For example, the property of alternating means that changing two different rows of an $n\times n$ matrix $M$ reverses the sign of $\det_n(M)$. If we view $\mathcal{M}_n(A)\simeq \text{End}_A(A^n)$, I wonder if we can give a more “global” characterization for $\det$ that does not involve specific elements of the ring or values of a function.

Moreover, determinants can be viewed as a natural transformation between two functors from $\textbf{CRing}$ to $\textbf{Grp}$. Can we them give another characterization for $\det$ in this case? What are the connections among all these characterizations?

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There is a very elegant description of determinants using exterior powers. Namely, if $V$ is a free module of rank $n$ and $T:V\to V$ is an endomorphism, then it induces an endomorphism $\bigwedge^n T:\bigwedge^n V\to \bigwedge^n V$. But $\bigwedge^n V$ is a free module of rank $1$, so any endomorphism of it is multiplication by some scalar. That scalar is the determinant of $T$.

(This definition makes many of the basic properties of the determinant trivial; for instance, its basis-invariance, and $\det(AB)=\det(A)\det(B)$. However, you don't actually get to escape doing the hard work of constructing the determinant with its basic properties: the fact that $\bigwedge^n V$ is free of rank $1$ is nontrivial and to prove it you basically need to construct an $n$-linear alternating function which sends the identity matrix to $1$.)

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