The first-order sensitivity matrices (matrices of sensitivity or influence coefficients) have application in many areas of water distribution system analysis. Finite-difference approximations, automatic differentiation, sensitivity equations, and the adjoint method have been used in the past to estimate sensitivity. In this paper new, explicit formulas for the first-order sensitivities of water distribution system (WDS) steady-state heads and flows to changes in demands, resistance factors, roughnesses, relative roughnesses, and diameters are presented. The formulas cover both pressure-dependent modeling (PDM) and demand-dependent modeling (DDM) problems in which either the Hazen-Williams or the Darcy-Weisbach head-loss models are used. Two important applications of sensitivity matrices, namely calibration and sensor placement, are discussed and illustrative examples of the use of sensitivity matrices in those applications are given. The use of sensitivity matrices in first-order confidence estimation is briefly discussed. The superior stability of the PDM formulation over DDM is established by the examination of the sensitivity matrices for the same network solved by both model paradigms. The sensitivity matrices and the key matrices in both the global gradient method for DDM problems and its counterpart for PDM problems have many elements in common. This means that the sensitivity matrices can be computed at marginal cost during the solution process with either of these methods.