I. The author gives, for every positive integer \(j\), necessary and sufficient conditions for a \(2\times 2\) real matrix (with positive determinant) to be a product of \(j\) positive definite real symmetric matrices (if \(j\ge 5\), every real matrix with positive determinant can be written as such a product). The author presents his results in a rather technical geometric context [see also J. Anal. Math. 12, 257--275 (1964; Zbl 0199.05502)]. \bigskip II. Given positive integers \(n\) and \(j\), which \(n\times n\) matrices (with positive determinant) can be written as a product of \(j\) positive definite matrices? The author first solves completely this question for complex \(2\times 2\) matrices (for the real case see Part I). He then shows that, for every \(n\), every \(n\times n\) real matrix with positive determinant can be written as a product of six positive definite real symmetric matrices, and every \(n\times n\) complex matrix with positive determinant can be written as a product of eleven positive definite Hermitian matrices. In later papers [J. Algebra 10, 174--182 (1968; Zbl 0225.15012), and Linear Algebra Appl. 3, 79--114 (1970; Zbl 0192.37002)] he completely answers the question raised above.