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Ex 4.5, 12 - Verify (AB)-1 = B-1 A-1, A = [3 7 2 5], B =

Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 2
Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 3 Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 4 Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 5 Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 6 Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 7 Ex 4.5, 12 - Chapter 4 Class 12 Determinants - Part 8

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Transcript

Ex 4.5, 12 Let A = [■8(3&[email protected]&5)] and B = [■8(6&[email protected]&9)] verify that (AB)-1 = B-1 A-1 Taking L.H.S (AB)–1 First calculating AB AB = [■8(3&[email protected]&5)] [■8(6&[email protected]&9)] = [■8(3(6)+7(7)&3(8)+7(9)@2(6)+5(7)&2(8)+5(9))] = [■8(18+49&[email protected]+35&16+45)] = [■8(67&[email protected]&61)] Now, (AB)-1 = 1/(|AB|) adj (AB) exists if |AB| ≠ 0 |AB| = |■8(67&[email protected]&61)| = 67 (61) – 47(87) = 4087 – 4089 = –2 Since |AB| ≠ 0 ∴ (AB)–1 exists Now, AB = [■8(67&[email protected]&61)] adj (AB) = [■8(67&[email protected]&61)] = [■8(61&−[email protected]−47&67)] Thus, (AB)–1 = 1/(|AB|) adj (AB) Putting values = 1/(−2) [■8(61&−[email protected]−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&[email protected]&9)| = 6(9) – 7(8) = 54 – 56 = –2 Since |B|≠ 0 ∴ B–1 exists Now, B = [■8(6&[email protected]&9)] adj B = [■8(6&[email protected]&9)] = [■8(9&−[email protected]−7&6)] Thus, B–1 = 1/(|B|) adj (B) = 1/(−2) [■8(9&−[email protected]−7&6)] Calculating A-1 A-1 = 1/(|A|) adj (A) exist if |A| ≠ 0 |A| = |■8(3&[email protected]&5)| = 15 – 14 = 1 Since |A| ≠ 0, A-1 exists A = [■8(3&[email protected]&5)] adj A = [■8(3&[email protected]&5)] = [■8(5&−[email protected]−2&3)] So, A–1 = 1/(|A|) adj (A) = 1/1 [■8(5&−[email protected]−2&3)] = [■8(5&−[email protected]−2&3)] Now B-1 A-1 = (−1)/2 [■8(9&−[email protected]−7&6)] [■8(5&−[email protected]−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−[email protected]−35−12&49+18)]

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.