A binomial-like matrix equation

International audienceWe show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. This is a generalization of Problem 4 (11th grade) from the Romanian National Mathematical Olympiad 2011

On nilpotent matrices of order 2

U ovom radu dajemo nekoliko karakterizacija kompleksnih nilpotentnih matrica reda 2, a koje se temelje na Hamilton–Cayleyjevom teoremu. U terminima tragova i determinanti matrica, opisujemo uvjete na matrice ( A,Binmathbb{M}_2(mathbb{C}) ) uz koje je komutator AB − BA nilpotentna matrica.In this paper, we give several characterizations of complex nilpotent matrices of order 2, which are based on the Cayley–Hamilton theorem. In terms of traces and determinants of matrices, we describe the conditions on ( A,Binmathbb{M}_2(mathbb{C}) ) under which the commutator AB − BA is a nilpotent matrix