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We describe a new Chebyshev spectral collocation method for systems of variable-coefficient linear delay differential equations with a single fixed delay. Computable uniform a posteriori bounds are given for this method. When the coefficients are periodic, the system has a unique compact nonnormal monodromy operator whose spectrum determines the stability of the system. The spectral method approximates this operator by a dense matrix of modest size. In cases where the coefficients are smooth we observe spectral convergence of the eigenvalues of that matrix to those of the operator. Our main result is a computable a posteriori bound on the eigenvalue approximation error in the case that the coefficients are analytic.