Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 10 / a / Solution

A finite-dimensional representation of a group $G$ on a vector space $V$ is a group homomorphism $D:G\to \mathrm{GL}(V)$. Thus each $D(X)$ is invertible andHere the group identity and the identity linear map are different objects, represented by corresponding symbols. Applying the multiplication property to $X{X}^{-1}=X^{-1}X=I$ givesBy the defining property of a matrix inverse,If the definition only states the multiplication law together with invertibility, $D(I{)}^2=D(I)$ already forces $D(I)=I_V$ after multiplication by its inverse.