Model identifiability is closely related to the completeness of Q -matrix. A Q -matrix is said to be complete if it can differentiate all latent classes. It is well known that for a Q -matrix to be complete, it must have a K by K identity submatrix. [5] However, having an identity is not sufficient enough. In our corollary 1 above, we require three identity matrices structure in Q -matrix to ensure a stronger sufficient conditions for identifiability of parameters. The following theorem generalizes the results to the cases of multi-category responses. Theorem 2 We consider the general setting of a latent class model with M > 2 classes. The re- sponse to item j takes k$_{j}$ possible values { 1 , ..., k$_{j}$ } . Let p k j α = P ( Y$_{j}$ = k | α ) and p$_{j}$ α = ( p 1 j $_{α}$, ..., p k$_{j}$ j $_{α}$) be the response vector. Let π α be the probability of each latent class. Suppose that the following conditions are satisfied for each item j . for some k = 1 , ..., k$_{j}$ − 1 . Then, the item parameters p k j α and the latent class population π α are identifiable up to a permuta- tion of the class labels. We now proceed to the results of the general diagnostic classification models in which the responses are multi-categorical variables and the response function may take more than two possible values. We need to introduce the T -matrix. Consider a test of K attributes and J items. The response to item j takes k$_{j}$ different values { 1 , ..., k$_{j}$ } and attribute i takes d$_{i}$ possible different values { 1 , ..., d$_{i}$ } . There are κ = ∏ J j $_{=1}$k$_{j}$ possible response patterns and M = ∏ K k $_{=1}$d$_{k}$ latent 12 B1 The response vector p$_{j}$ α takes only two distinct values for all α = 1 , ..., M . B2 There exist three non-overlap subsets of items denoted by I$_{1}$ , I$_{2}$ , and I$_{3}$ such that for α 1 = α 2 and l = 1 , 2 , and 3 , there exists j ∈ I$_{l}$ such that p 1 j $^{α}$1 + ... + p k j $^{α}$1 = p 1 j $^{α}$2 + ... + p k j $^{α}$2 B3 π α > 0 for α = 1 , 2 , . . . , M .