Due to its ability to significantly improve data rate, intelligent reflecting surface (IRS) is a potential crucial technique for the future generation wireless networks like six-generation (6G). In this paper, we focus on the analysis of degree of freedom (DoF) in IRS-aided multi-user multi-input multi-output (MIMO) network. Firstly, the DoF upper bound of IRS-aided single-user MIMO network, i.e., the achievable maximum DoF of such a system, is derived, and the corresponding results are extended to the case of IRS-aided multiuser MIMO by using the matrix rank inequalities. In particular, in serious rank-deficient, also called low-rank, like line-of-sight channel, the network DoF may double over no-IRS with the help of IRS. To verify the rate performance gain from augmented DoF, three closed-form beamforming methods, null-space projection plus maximize transmit power and maximize receive power (NSP-MTP-MRP), Schmidt orthogonalization plus (SO-MMSE) and two-layer leakage plus minimum mean square error (TLL-MMSE) are proposed to achieve the maximum DoF. Simulation results show that IRS does make a dramatic rate enhancement. For example, in a serious rank-deficient channel, also called low-rank, the sum-rate of the proposed TLL-MMSE aided by IRS is up to 2.54 times that of no IRS. This means that IRS may make a significant DoF improvement in such a channel.