Past exam of the mathematical methods I course of the University of Oxford / 2015 / 4 / Solution

The addition formulas for the hyperbolic functions implySince $M=B(y)/2$, its fourth matrix power isAlternatively, the orthonormal eigenvectors $(1,\pm 1)/\sqrt{2}$ have eigenvalues $e^{\pm y}/2$. Raising these eigenvalues to the fourth power gives the same result by spectral decomposition.