The cross section for the production of single top quarks in the $t$ channel is measured in proton-proton collisions at 13 TeV with the CMS detector at the LHC. The analyzed data correspond to an integrated luminosity of 2.3 fb$^{-1}$. The event selection requires one muon and two jets where one of the jets is identified as originating from a bottom quark. Several kinematic variables are then combined into a multivariate discriminator to distinguish signal from background events. A fit to the distribution of the discriminating variable yields a total cross section of 232 $\pm$ 13 (stat) $\pm$ 28 (syst) pb and a ratio of top quark and top antiquark production of $R_{t\textrm{-ch.}}= $ 1.81 $\pm$ 0.18 (stat) $\pm$ 0.15 (syst). From the total cross section the absolute value of the CKM matrix element $V_{\mathrm{tb}}$ is calculated to be 1.03 $\pm$ 0.07 (exp) $\pm$ 0.02 (theo). All results are in agreement with the standard model predictions.