Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. The set D_L of tuples X=(X_1,...,X_g) of symmetric matrices (of the same size) for which L(X):=I-\sum A_j \otimes X_j is PsD, is a matrix convex set, called a free spectrahedron. We explain that any tuple X of symmetric matrices in a free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron S_L. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive. In the case when S_L is the hypercube [-1,1]^g, we derive an analytical formula for this scale factor, which as a by-product gives new probabilistic results for the binomial and beta distributions.