The concept of central polynomials was first introduced by \textit{E. Formanek} [J. Algebra 23, 129-132 (1972; Zbl 0242.15004)], and by \textit{Yu. P. Razmyslov} [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 483-501 (1973; Zbl 0314.16016)]. The central polynomials play a key role in the structure of the matrix \(T\)-ideals. The first central polynomials for the matrix algebras \(M_n(K)\), \(\text{char }K=0\) were of degree \(n^2\). Then \textit{V. Drensky} and \textit{A. Kasparian} [Commun. Algebra 13, 745-752 (1985; Zbl 0556.16007)] proved that the minimal central polynomial for \(M_3(K)\) is of degree 8. In addition, \textit{V. Drensky} and \textit{G. M. Piacentini Cattaneo} [J. Algebra 168, No. 2, 469-478 (1994; Zbl 0834.16020)] showed that there exists a central polynomial for \(M_4(K)\) whose degree equals 13. The main result of the paper under review consists of constructing a central polynomial for \(M_n(K)\), \(\text{char }K=0\) whose degree equals \((n-1)^2+4\), \(n\geq 3\). In order to construct this polynomial the author makes use of a certain weak polynomial identity for \(M_n(K)\) (i.e. an associative polynomial that vanishes on \(sl_n(K)\) but that does not vanish on \(M_n(K)\)). These central polynomials are of the least known degree. Note that the paper could be an important step to finding the central polynomials of minimal degrees.