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Misc 6 - Find x, y, z if the matrix A satisfies A'A = I - Miscellaneou

Misc. 6 - Chapter 3 Class 12 Matrices - Part 2
Misc. 6 - Chapter 3 Class 12 Matrices - Part 3

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Misc 6 Find the values of x, y, z if the matrix A = [■8(0&2𝑩&𝑧@đ‘„&𝑩&−𝑧@đ‘„&−𝑩&𝑧)] satisfy the equation Aâ€ČA = I. Given, A = [■8(0&2𝑩&𝑧@đ‘„&𝑩&−𝑧@đ‘„&−𝑩&𝑧)] A’ = [■8(0&đ‘„&đ‘„@2𝑩&𝑩&−𝑩@𝑧&−𝑧&𝑧)] I = [■8(1&0&0@0&1&0@0&0&1)] Now, A’A = I Putting values [■8(0&đ‘„&đ‘„@2𝑩&𝑩&−𝑩@𝑧&−𝑧&𝑧)][■8(0&2𝑩&𝑧@đ‘„&𝑩&−𝑧@đ‘„&−𝑩&𝑧)] = [■8(1&0&0@0&1&0@0&0&1)] [■8(0(0)+đ‘„(đ‘„)+đ‘„(đ‘„)&0(2𝑩)+đ‘„(𝑩)+đ‘„(−𝑩)&0(𝑧)+đ‘„(−𝑧)+đ‘„(𝑧)@2𝑩(0)+𝑩(đ‘„)−𝑩(đ‘„)&2𝑩(2𝑩)+𝑩(𝑩)−𝑩(−𝑩)&2𝑩(𝑧)+𝑩(−𝑧)−𝑩(𝑧)@𝑧(0)−𝑧(đ‘„)+𝑧(đ‘„)&𝑧(2𝑩)−𝑧(𝑩)+𝑧(−𝑩)&𝑧(𝑧)−𝑧(−𝑧)+𝑧(𝑧))] = [■8(1&0&0@0&1&0@0&0&1)] [■8(0+đ‘„^2+đ‘„^2&0+đ‘„đ‘Šâˆ’đ‘„đ‘Š&0âˆ’đ‘„đ‘§+đ‘„đ‘§@0+đ‘„đ‘Šâˆ’đ‘„đ‘Š&4𝑩^2+𝑩^2+𝑩^2&2𝑧𝑩−𝑧𝑩−𝑧𝑩@0âˆ’đ‘„đ‘§+đ‘„đ‘§&2𝑧𝑩−𝑧𝑩−𝑧𝑩&𝑧^2+𝑧^2+𝑧^2 )]= [■8(1&0&0@0&1&0@0&0&1)] [■8(2đ‘„^2&0&0@0&6𝑩^2&0@0&0&3𝑧^2 )]= [■8(1&0&0@0&1&0@0&0&1)] Since matrices are equal, corresponding elements are equal Thus, x = ± 1/√2 , y = ± 1/√6 , z = ± 1/√3 2x2 = 1 x2 = 1/2 x = ±√(1/2) x = ± 𝟏/√𝟐 6y2 = 1 y2 = 1/6 y = ±√(1/6) y = ± 𝟏/√𝟔 3z2 = 1 z2 = 1/3 z = ±√(1/3) z = ± 𝟏/√𝟑

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.