Introduction. The class of Minkowski matrices consists of square matrices of the form a -a, where a is the identity matrix and a, with real or complex elements, satisfies the condition (1). The inequality is given in the lemma, which improves the author's previous result [3, p. 239] by removing two restrictions.2 Refinements of the inequality are given in ?3. G. B. Price [2] and A. M. Ostrowski [1] give bounds for determinants with dominant principal diagonal. It loses no generality to consider the square matrices with units on the principal diagonal. Thus our results may be applied to the determinants studied by Price and Ostrowski. We apply the inequality of our lemma to obtain boUinds for the determinant of a-a. Our results in (9) and (15) are better estimates than those of Price and Ostrowski. The main idea of our method centers on (13) and (14). The concept of quasi-inverse, which was used in [3], is no longer needed. We use the notation a(i, j) instead of ais.