Linear Algebra Done Right · Problem 9C.21

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

n

is a positive integer and

\delta\colon \mathbf{C}^{n,n} \to \mathbf{C}

is a function such that

\delta(AB) = \delta(A) \cdot \delta(B)

for all

A, B \in \mathbf{C}^{n,n}

and

\delta(A)

equals the product of the diagonal entries of

A

for each diagonal matrix

A \in \mathbf{C}^{n,n}

. Prove that

\delta(A) = \det A

for all

A \in \mathbf{C}^{n,n}

. Recall that

\mathbf{C}^{n,n}

denotes the set of

n

-by-

n

matrices with entries in

\mathbf{C}

. This exercise shows that the determinant is the unique function defined on square matrices that is multiplicative and has the desired behavior on diagonal matrices. This result is analogous to Exercise 10 in Section 8D, which shows that the trace is uniquely determined by its algebraic properties.

Related theory

determinants

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