Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 14 / b / ii

Insert the eigenvalues and eigenvectors into the inverse spectral decomposition. The first two projections contributeAdding the third projection givesMultiplication by $A$ gives the identity; in particular the upper block satisfies $\left(\begin{array}{cc}3& -1\\ -1& 3\end{array}\right)\left(\begin{array}{cc}3& 1\\ 1& 3\end{array}\right)=8I$.