In this paper we extend the classical Bohr’s inequality to the setting of the non-commutative Hardy space H^1 associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr’s inequality for operators in the von Neumann-Schatten class \mathcal C_1 and square matrices of any finite order. Interestingly, we establish that the optimal bound for r in the above mentioned Bohr’s inequality concerning von Neumann-Schatten class is 1/3 whereas it is 1/2 in the case of 2\times 2 matrices and reduces to \sqrt {2}-1 for the case of 3\times 3 matrices. We also obtain a generalization of our above-mentioned Bohr’s inequality for finite matrices where we show that the optimal bound for r, unlike above, remains 1/3 for every fixed order n\times n,\ n\ge 2.