Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 4 / a / iii / Solution

The definition of an eigenvalue and the kernel criterion for a singular square matrix giveFor the second equivalence, a linear map from a finite-dimensional vector space to itself has zero kernel exactly when its rank of a linear map is the full dimension, by the rank-nullity theorem, and therefore exactly when it is invertible. This proves both directions.