Ex 3.3, 6 (ii) - Chapter 3 Class 12 Matrices

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Ex 3.3, 6 (ii) If A = [■8(sin⁡𝛼&cos⁡𝛼@−cos⁡𝛼&sin⁡𝛼 )] , then verify that A’ A = I Solving L.H.S A’ A Given A = [■8(sin⁡𝛼&cos⁡𝛼@−cos⁡𝛼&sin⁡𝛼 )] So, A’ = [■8(sin⁡𝛼&〖−cos〗⁡𝛼@cos⁡𝛼&sin⁡𝛼 )] A’ A = [■8(sin⁡𝛼&〖−cos〗⁡𝛼@cos⁡𝛼&sin⁡𝛼 )] [■8(sin⁡𝛼&cos⁡𝛼@〖−cos〗⁡𝛼&sin⁡𝛼 )] = [■8(sin⁡𝛼 〖.sin〗⁡𝛼+〖(−cos〗⁡〖𝛼)〖(−cos〗⁡〖𝛼)〗 〗&sin⁡𝛼 〖.cos〗⁡𝛼+〖(−cos〗⁡〖𝛼)〖(sin〗⁡〖𝛼)〗 〗@cos⁡𝛼 〖.sin〗⁡𝛼+sin⁡〖𝛼 〖(−cos〗⁡〖𝛼)〗 〗&cos⁡𝛼 〖.cos〗⁡𝛼+sin⁡〖𝛼 〖.sin〗⁡𝛼 〗 )] = [■8(sin2⁡𝛼+cos2𝛼&sin⁡〖𝛼 cos⁡〖𝛼−cos⁡〖𝛼 sin⁡𝛼 〗 〗 〗@cos⁡𝛼 sin⁡〖𝛼−sin⁡𝛼 〗 cos𝛼&cos2⁡𝛼+sin2𝛼)] = [■8(𝐬𝐢𝐧𝟐⁡𝜶+𝐜𝐨𝐬𝟐𝜶&𝟎@𝟎&𝐜𝐨𝐬𝟐⁡𝜶+𝐬𝐢𝐧𝟐𝜶)] Using sin2 θ + cos2 θ = 1 = [■8(1&0@0&1)] = I = R.H.S Hence L.H.S = R.H.S Hence Proved