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In this paper we prove the Hardy inequalities for the quadratic form of theLaplacian with the Landau Hamiltonian magnetic field. Moreover, we obtainPoincar\'e type inequality and inequalities with more general families ofweights, all with estimates for the remainder terms of these inequalities.Furthermore, we establish weighted Hardy inequalities for the quadratic form ofthe magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohmtype. For these, we show refinements of the known Hardy inequalities for theBaouendi-Grushin operator involving radial derivatives in some of thevariables. The corresponding uncertainty type principles are also obtained.

Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I