A quantum observable is represented by the matrix with rows (2, 1) and (1, 2). Its possible measured values are the eigenvalues of that matrix. What is the LARGER eigenvalue?

3

$(2-\lambda)^{2} - 1 = 0$ gives $\lambda = 1$ or $\lambda = 3$, so the larger is 3. This is the link the lesson insists on: the measurable energies of a quantum system are not merely calculated with linear algebra, they are the eigenvalues of an operator.