Idempotency of linear combinations of three idempotent matrices, two of which are disjoint Original Research Article

Given nonzero idempotent matrices A1,A2,A3 such that A2 and A3 are disjoint, i.e., A2A3=0=A3A2, the problem of characterizing all situations, in which a linear combination C=c1A1+c2A2+c3A3 is an idempotent matrix, is studied. The results obtained cover those established by J.K. Baksalary, O.M. Baksalary, and G.P.H. Styan (Linear Algebra Appl. 354 (2002) 21) under the additional assumption that c3=−c2, i.e., in the particular case where C=c1A1+c2(A2−A3) is actually a linear combination of an idempotent matrix A1 and a tripotent matrix A2−A3.