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C3091511 - A 11 C A 12 A-1 22 A-1 22 A 22 = ± I n I n ...

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11. (Statement of the problem.) Proof. Suppose that A 11 and A 22 are nonsingular. By definition, A - 1 11 and A - 1 22 exist. ( Therefore it is legal to use A - 1 11 and A - 1 22 . ) Take an n × n matrix C = - A - 1 11 A 12 A - 1 22 . So A - 1 11 A 12 + CA 22 = A - 1 11 A 12 - A - 1 11 A 12 A - 1 22 A 22 = A - 1 11 A 12 - A - 1 11 A 12 I = 0 , A 11 C + A 12 A - 1 22 = - A 11 A - 1 11 A 12 A - 1 22 + A 12 A - 1 22 = - IA 12 A - 1 22 + A 12 A - 1 22 = 0 . Hence we have ± A - 1 11 C 0 A - 1 22 ¶± A 11 A 12 0 A 22 = ± A - 1 11 A 11 A - 1 11 A 12 + CA 22 0 A - 1 22 A 22 = ± I n 0 0 I n = I 2 n , ± A 11 A 12 0 A 22 ¶± A - 1 11 C 0 A - 1 22 = ± A 11 A - 1 11
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Unformatted text preview: A 11 C + A 12 A-1 22 A-1 22 A 22 ¶ = ± I n I n ¶ = I 2 n . (a) By definition and the above two equations, A is nonsingular and A-1 has the form ± A-1 11 C A-1 22 ¶ . (b) As we stated, C =-A-1 11 A 12 A-1 22 . / 1...
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