An elementary proof is given of the fact that an n–tuple A = (A₁, …, A_n) of self-adjoint matrices in a 2-dimensional Hilbert space consists of mutually commuting matrices A_j, 1 ≤ j ≤ n, if and only if γ(A) is non-empty. Here γ(A) ⊆ ℝⁿ is the joint spectrum of A (in the sense of McIntosh and Pryde) consisting of those points β ∈ ℝⁿ for which the matrix is not invertible.