Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 7B / c

Consider the real polynomialThe hypothesis says $p(i)\ne 0$, so $p$ is not identically zero. A nonzero polynomial has only finitely many roots, whereas $\mathbb{R}$ is infinite. Hence there is a real $\lambda$ withThus $A+\lambda B$ is invertible, as summarized by the real parameter avoiding a singular matrix pencil.