A completion problem over the field of real numbers

Let F be a field. Almost. 25 years ago, G.N. de Oliveira has proposed the f ollowing completion problems: Describe the possible characteristic polynomi als of [A(i,j)], i, j is an element of {1, 2}, where A(1,1) and A(2,2) are square submatrices, when some of the blocks A(i,j) are fixed and the others vary [cf, Linear Multilinear Algebra 2 (1975) 357]. Several of these probl ems remain unsolved. This paper gives the solution, over the field of real numbers, of Oliveira's problem where the blocks A(1,2), A(2,1) are fixed an d the others vary. (C) 2000 Elsevier Science Inc. All rights reserved. AMS classification: 15A18.