If A is a symmetric and B skew symmetric matrix and (A + B) is non-singular and C = (A + B) –1 (A – B), then prove that
(i)C T (A + B) C = A + B
(ii)C T (A – B) C = A – B
Text Solution
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(i)(A + B) C = (A +B) (A + B) –1 (A – B)
(A + B) C = A – B .........(i)
C T = ((A + B) –1 (A – B)) T = (A – B) T ((A + B) –1 ) T
= (A – B) T ((A + B) T ) –1 Here {|A + B| ≠ 0 ⇒ (A + B) T | ≠ 0 ⇒ |A – B | ≠ 0 }
= (A + B) (A – B) –1 ........(ii)
By (i) and (ii)
C T (A + B) C = (A + B) (A – B) –1 (A – B)
C –1 (A + B) C = A + B ........(iii)
(ii) Taking transpose of (iii)
C T (A – B) C = A – B.
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