Ex 4.4, 12 - Verify (AB)-1 = B-1 A-1, A = [3 7 2 5], B = - Ex 4.4

part 2 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants
part 3 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants
part 4 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants
part 5 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants part 6 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants part 7 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants part 8 - Ex 4.4, 12 - Ex 4.4 - Serial order wise - Chapter 4 Class 12 Determinants

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Ex 4.4, 12 Let A = [■8(3&7@2&5)] and B = [■8(6&8@7&9)] verify that (AB)-1 = B-1 A-1 Taking L.H.S (AB)–1 First calculating AB AB = [■8(3&7@2&5)] [■8(6&8@7&9)] = [■8(3(6)+7(7)&3(8)+7(9)@2(6)+5(7)&2(8)+5(9))] = [■8(18+49&24+63@12+35&16+45)] = [■8(67&87@47&61)] Now, (AB)-1 = 1/(|AB|) adj (AB) exists if |AB| ≠ 0 |AB| = |■8(67&87@47&61)| = 67 (61) – 47(87) = 4087 – 4089 = –2 Since |AB| ≠ 0 ∴ (AB)–1 exists Now, AB = [■8(67&87@47&61)] adj (AB) = [■8(67&87@47&61)] = [■8(61&−87@−47&67)] Thus, (AB)–1 = 1/(|AB|) adj (AB) Putting values = 1/(−2) [■8(61&−87@−47&67)] Taking R.H.S B-1A-1 First Calculating B-1 B–1 = 1/(|B|) adj (B) exist if |B|≠ 0 Now, |B| = |■8(6&8@7&9)| = 6(9) – 7(8) = 54 – 56 = –2 Since |B|≠ 0 ∴ B–1 exists Now, B = [■8(6&8@7&9)] adj B = [■8(6&8@7&9)] = [■8(9&−8@−7&6)] Thus, B–1 = 1/(|B|) adj (B) = 1/(−2) [■8(9&−8@−7&6)] Calculating A-1 A-1 = 1/(|A|) adj (A) exist if |A| ≠ 0 |A| = |■8(3&7@2&5)| = 15 – 14 = 1 Since |A| ≠ 0, A-1 exists A = [■8(3&7@2&5)] adj A = [■8(3&7@2&5)] = [■8(5&−7@−2&3)] So, A–1 = 1/(|A|) adj (A) = 1/1 [■8(5&−7@−2&3)] = [■8(5&−7@−2&3)] Now B-1 A-1 = (−1)/2 [■8(9&−8@−7&6)] [■8(5&−7@−2&3)] = (−1)/2 [■8(9(5)+( –8)( –2)&9(−7)+(−8)(3)@ –7(5)+6( –2)&−7(−7)+6(3))] = (−1)/2 [■8(45+16&−63−24@−35−12&49+18)] = (−1)/2 [■8(61&−87@−47&67)] = L.H.S ∴ L.H.S = R.H.S Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo