  Encouraged by a hint in a search for right-handed $W$ bosons at the LHC, we
investigate whether the unitarity of a right-handed quark mixing matrix and the
equality of the left- and right-handed quark mixing matrices could be tested at
the LHC. We propose a particular test, involving counting the numbers of
$b$-tags in the final state, and simulate the test at the event level with
Monte-Carlo tools for the forthcoming $\sqrt{s}=13$ TeV LHC run. We find that
testing unitarity with 20/fb will be challenging; our test successfully rejects
unitarity if the right-handed quark mixing matrix is non-unitary, but only in
particular cases. On the other hand, our test may provide the first opportunity
to test the unitarity of a right-handed quark mixing matrix and with 3000/fb
severely constrains possible departures from unitarity in the latter. We refine
our previous work, testing the equality of quark mixing matrices, with full
collider simulation. With 20/fb, we are sensitive to mixing angles as small as
$30^\circ$, and with 3000/fb, angles as small as $7.5^\circ$, confirming our
preliminary analysis. We briefly investigate testing the unitarity of the SM
CKM matrix with a similar method by studying semileptonic $t\bar t$ production,
concluding that systematics make it particularly difficult.
