The sets of 2(2s + 1)-component irreducible and Clifford algebraic Hermitian and unitary matrices through the two-component Pauli matrices are suggested, where s = 1/2, 3/2, 5/2,.... Using these matrix sets, the eigenvalues of which are +/- 1, the 2(2s + 1)-component generalized Dirac equation for a description of arbitrary half-integral spin particles is constructed. In accordance with the correspondence principle, the generalized Dirac equation suggested arises from the condition of relativistic invariance. This equation is reduced to the sets of two-component matrix equations the number of which is equal to 2s + 1. The new relativistic invariant equation of motion leads to an equation of continuity with a positive-definite probability density and also to the Klein-Gordon equation. This relativistic equation is causal in the presence of an external electromagnetic field interaction. It is shown that, in the case of nonrelativistic limit, the relativistic equation presented is reduced to the Pauli equation describing the motion of half-integral spin particle in the electromagnetic field.