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Ex 4.4, 11 Find the inverse of each of the matrices [■8(1&0&0@0&cos⁡𝛼&sin⁡𝛼@0&sin⁡𝛼&−cos⁡𝛼 )] Let A =[■8(1&0&0@0&cos⁡𝛼&sin⁡𝛼@0&sin⁡𝛼&−cos⁡𝛼 )] We know that A–1 = 1/(|A|) (adj A) exists if |A|≠ 0 Calculating |A| |A| = |■8(1&0&0@0&cos⁡𝛼&sin⁡𝛼@0&sin⁡𝛼&−cos⁡𝛼 )| = 1 |■8(cos⁡𝛼&sin⁡𝛼@sin⁡𝛼&−cos⁡𝛼 )|– 0 |■8(0&sin⁡𝛼@0&〖− cos〗⁡𝛼 )|+ 0|■8(0&cos⁡𝛼@0&𝑠𝑖𝑛 𝛼)| = 1(– cos2α – sin2α ) – 0 + 0 = –( cos2α + sin2α ) = –1 Since |𝐴|≠ 0 Thus A-1 exists Calculating adj A adj (A) = [■8(A11&A21&A31@A12&A22&A32@A13&A23&A33)] A = [■8(1&0&0@0&cos⁡𝛼&sin⁡𝛼@0&sin⁡𝛼&−cos⁡𝛼 )] M11 = |■8(cos⁡"α" &sin⁡"α" @sin⁡"α" &−cos⁡"α" )| = –cos2α – sin2α = –(cos^2α 〖+ 𝑠𝑖𝑛〗^2α ) = –1 M12 = |■8(0&sin 𝛼@0&−cos 𝛼)| = 0 – 0 = 0 M13 = |■8(0&cos⁡𝛼@0&sin 𝛼)| = 0 – 0 = 0 M21 = |■8(0&0@sin 𝛼&−cos⁡𝛼 )| = 0 – 0 = 0 M22 = |■8(1&0@0&−cos 𝛼)| = –cos α – 0 = –cos α M23 = |■8(1&0@0&sin⁡𝛼 )| = sin α = 0 = sin α M31 = |■8(0&0@cos 𝛼&sin 𝛼)| = 0 – 0 = –0 M32 = |■8(1&0@0&sin 𝛼)| = sin α – 0 = sin α M33 = |■8(1&0@0&cos 𝛼)| = cos α + 0 = cos α Now, A11 = 〖(−1)〗^(1+1) M11 = 〖(−1)〗^2 (–1)2 = –1 A12 = 〖(−1)〗^(1+2) M12 = 〖(−1)〗^3 0 = 0 A13 = 〖(−1)〗^(1+3) M13 = 〖(−1)〗^4 0 = 0 A21 = 〖(−1)〗^(2+1)M21 = (–1)3 0 = 0 A22 = 〖(−1)〗^(2+2) M22 = 〖(−1)〗^4(– cos α) = –cos α A23 = 〖"(– 1)" 〗^(2+3) M23 = 〖"(–1)" 〗^5 sin α = –sin α A31 = 〖(−1)〗^(3+1). M31 = 〖(−1)〗^4 0 = 0 A32 = 〖(−1)〗^(3+2)sin α = (–1)5 sin α = –sin α A33 = 〖(−1)〗^(3+3)M33 = (–1)6 cos α = cos α So, adj (A) = [■8(A11&A21&A31@A12&A22&A32@A33&A23&A33)] = [■8(−1&0&0@0&−cos⁡𝛼&−sin⁡𝛼@0&−sin⁡𝛼&cos⁡𝛼 )] Calculating inverse Now, A– 1 = 1/(|A|) ( adj (A)) = 1/(−1) [■8(−1&0&0@0&−cos⁡𝛼&−sin⁡𝛼@0&−sin⁡𝛼&cos⁡𝛼 )] = – [■8(−1&0&0@0&−cos⁡𝛼&−sin⁡𝛼@0&−sin⁡𝛼&cos⁡𝛼 )] = [■8(𝟏&𝟎&𝟎@𝟎&𝒄𝒐𝒔⁡𝜶&𝒔𝒊𝒏⁡𝜶@𝟎&𝒔𝒊𝒏⁡𝜶&〖−𝒄𝒐𝒔〗⁡𝜶 )]

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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