Derive eigenvalues and sum for inverse matrix
Company: DRW
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Onsite
Let A be an invertible n×n matrix with eigenvalues {λ1, …, λn} (all nonzero). Prove that the eigenvalues of A^{-1} are {1/λ1, …, 1/λn}. Then compute the sum of the eigenvalues of A^{-1} and express it in terms of A (e.g., as tr(A^{-1})); discuss any assumptions needed for these equalities to hold.
Quick Answer: This question evaluates understanding of matrix spectral properties, specifically the relationship between the eigenvalues of an invertible matrix and those of its inverse, plus familiarity with matrix invariants like the trace.