Let \({\mathcal A}\) be an Artinian algebra over a field F with basis \(A=\{a_ 1,...,a_ n\}\), and let \(\lambda\) and \(\rho\) be left and right regular representations of \({\mathcal A}\) respectively. Denote by tr(x) the trace of a square matrix x over F. Put \(\lambda_{ij}=tr(\lambda (a_ ia_ j))\), \(\rho_{ij}=tr(\rho (a_ ia_ j))\), and let \(L^ A=(\lambda_{ij})\), \(R^ A=(\rho_{ij})\) be the corresponding \(n\times n\)-matrices over F (the author uses another but equivalent definition of \(L^ A\) and \(R^ A)\). In the paper several propositions and theorems are proved which show a connection between properties of \({\mathcal A}\) and those of \(L^ A\) and \(R^ A\). An algebra \({\mathcal A}\) is said to be dual iff \(L^ A=R^ A\) (it is shown that the definition does not depend on the choice of the basis A). Theorem 4 states that any Frobenius algebra is dual. The author asks whether the same is true for any quasi-Frobenius algebra; he notes that there exists a dual quasi-Frobenius algebra which is not Frobenius. The referee thinks that the calculating proofs of Propositions 1,5 and Theorem 4 given in the paper are unnecessary because these assertions are mere consequences of the definition of \(L^ A\) and \(R^ A\) and Nakayama's definition of Frobenius algebras as algebras for which \(\lambda\) and \(\rho\) are equivalent.