Derive eigenvalues and sum for inverse matrix

Q: Derive eigenvalues and sum for inverse matrix

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Question

Eigenvalues of an Inverse and Their Sum

Context

Let A be an invertible n×n matrix (over the real or complex numbers). All eigenvalues of A are nonzero because A is invertible.

Tasks

Prove that the eigenvalues of A^{-1} are {1/λ1, …, 1/λn}, where {λ1, …, λn} are the eigenvalues of A (counted with algebraic multiplicity).
Compute the sum of the eigenvalues of A^{-1} and express it in terms of A (e.g., as tr(A^{-1})).
State any assumptions needed for these equalities to hold.

Derive eigenvalues and sum for inverse matrix

Eigenvalues of an Inverse and Their Sum

Context

Tasks

Solution (Locked)

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