(a) Give an example to show that if A can be partitioned as

where P, Q, R, and S are all square, then it is not necessarily true that
det A = (det P) (det S) - (det Q) (det R)
(b) Assume that A is partitioned as in part (a) and that P is invertible. Let

Compute det (BA) using Exercise 69 and use the result to show that
det A = det P det(S - RP-1 Q)
[The matrix S - RP-1Q is called the Schur complement of P in A, after Issai Schur (1875- 1941), who was born in Belarus but spent most of his life in Germany. He is known mainly for his fundamental work on the representation theory of groups, but he also worked in number theory, analysis, and other areas.]
(c) Assume that A is partitioned as in part (a), that P is invertible, and that PR = RP. Prove that
detA = det(PS - RQ)

Transcribed Image Text:

RP-1 I