Question

if B and C are matices with ranks r and s, respectively and A = B 0, 0 C (2 x2 with B and C as diagonals), show that A is of rank r+s

Answer 1

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Answer 2

Take the matrix D = \[(B \ 0)\]. You know it is rank r, because B is rank r. Also you know matrix E = \[(0 \ C)\] is rank s, because C is rank s. Notice that the columns of D are perpendicular or orthogonal (zero dot products) to the columns of E. This must mean that the column space of D does only intersects the column space of E at 0. Therefore, matrix A (which has their joint column space) is rank r+s.