Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 6 / Solution

An invertible matrix is a square matrix $X$ admitting a matrix inverse $X^{-1}$ with $X{X}^{-1}=X^{-1}X=I$, where $I$ is the identity matrix. Equivalently, its associated linear map is bijective.

Let $X\mathbf{v}=\lambda \mathbf{v}$ with eigenvector $\mathbf{v}\ne 0$. The eigenvalue cannot be zero: otherwise applying $X^{-1}$ gives $\mathbf{v}=0$. Apply $X^{-1}$ to the eigenvector equation:Thus $1/\lambda$ is an eigenvalue of $X^{-1}$ with the same eigenvector.