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The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadamard and generalized Kronecker products, respectively. We define the Khatri-Rao and Tracy-Singh sums for partitioned matrices as generalized Hadamard and generalized Kronecker sums and derive some results including matrix equalities and inequalities involving the two sums. Based on the connection between the Khatri-Rao and Tracy-Singh products (sums) and use mainly Liu's, Mond and Pečarić's methods to establish new inequalities involving the Khatri-Rao product (sum). The results lead to inequalities involving Hadamard and Kronecker products (sums), as a special case.