Decomposition of matrices into commutators of reflections. (Q1873637)

It is proved that every \(n\times n\)-matrix \(A\) belonging to the group \(\text{GL}_n(F)\) defined over field \(F\) is the product of at most \([\operatorname {res} A/2]+2\) commutators of reflections [see \textit{E. Cartan} (1938), \textit{J. Dieudonne} (1948) and \textit{P. Scherk}, Proc. Am. Math. Soc. 1, 481--491 (1950; Zbl 0039.01004)] for all \(n\geq 2\), except for \(n=2\) and \(F=F_2\), where \([\;]\) denotes the integer part of a rational number and \(\operatorname {res}A= \text{rank}(A-I)\).