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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&[email protected]&1&[email protected]&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&[email protected]&1&[email protected]&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&[email protected]&1&[email protected]&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&[email protected]&1&[email protected]&0&1)] [■8(2đ‘„^2&0&[email protected]&6𝑩^2&[email protected]&0&3𝑧^2 )]= [■8(1&0&[email protected]&1&[email protected]&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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.