Misc 3 - Find x, y, z if the matrix A satisfies A'A = I - Miscellaneou - Miscellaneous

part 2 - Misc 3 - Miscellaneous - Serial order wise - Chapter 3 Class 12 Matrices
part 3 - Misc 3 - Miscellaneous - Serial order wise - Chapter 3 Class 12 Matrices

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Misc 3 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&𝒙&𝒙@𝟐𝒚&𝒚&−𝒚@𝒛&−𝒛&𝒛)] I = [■8(1&0&0@0&1&0@0&0&1)] Now, A’A = I Putting values [■8(𝟎&𝒙&𝒙@𝟐𝒚&𝒚&−𝒚@𝒛&−𝒛&𝒛)][■8(𝟎&𝟐𝒚&𝒛@𝒙&𝒚&−𝒛@𝒙&−𝒚&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(𝟐𝒙^𝟐&𝟎&𝟎@𝟎&𝟔𝒚^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

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