SEVERAL QUESTIONS OF LONGSTAFF ON GENERATION IN MATRIX ALGEBRAS

Abstract We solve three problems raised in recent articles by Longstaff [ Bull. Aust. Math. Soc. 102 (2020), 226–236; 103 (2021), 260–270; 104 (2021), 78–93]. We show that: (1) the length of an irreducible family of $n\times n$ rank-one matrices can take any value $1,2,\ldots ,n$ ; (2) the slot length of an irreducible pair of $n\times n$ matrices can exceed $2n-4$ ; and (3) we give bounds on the $1$ -minimum spanning lengths of irreducible pairs.