Following a paper of \textit{S. Liu} and \textit{H. Neudecker} [ibid. 197, No. 1, 23-26 (1996; Zbl 0853.15014)], the authors present sufficient and necessary conditions under which equality occurs in matrix Kantorovich-type inequalities. They also present several relevant inequalities. The following concluding comments are stated. (i) By using the results of Section 3, one can examine the special cases for the Hadamard product, and for the upper-left submatrices. (ii) Applying the black-method, plenty of results for several cases of sums of matrices including Kantorovich and Cauchy-Schwarz inequalities can be easily derived from the results in this paper. (iii) Only the case which involves one positive definite matrix is considered in Section 3.