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Putnam 1986 B6

Source:

August 5, 2019
Putnam

Problem Statement

Suppose A,B,C,DA,B,C,D are n×nn \times n matrices with entries in a field FF, satisfying the conditions that ABTAB^T and CDTCD^T are symmetric and ADT−BCT=IAD^T - BC^T = I. Here II is the n×nn \times n identity matrix, and if MM is an n×nn \times n matrix, MTM^T is its transpose. Prove that ATD−CTB=IA^T D - C^T B = I.
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