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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 = [β– 8(0&2𝑦&𝑧@π‘₯&𝑦&βˆ’π‘§@π‘₯&βˆ’π‘¦&𝑧)] satisfy the equation Aβ€²A = I. Given, A = [β– 8(0&2𝑦&𝑧@π‘₯&𝑦&βˆ’π‘§@π‘₯&βˆ’π‘¦&𝑧)] A’ = [β– 8(0&π‘₯&π‘₯@2𝑦&𝑦&βˆ’π‘¦@𝑧&βˆ’π‘§&𝑧)] I = [β– 8(1&0&0@0&1&0@0&0&1)] Now, A’A = I Putting values [β– 8(0&π‘₯&π‘₯@2𝑦&𝑦&βˆ’π‘¦@𝑧&βˆ’π‘§&𝑧)][β– 8(0&2𝑦&𝑧@π‘₯&𝑦&βˆ’π‘§@π‘₯&βˆ’π‘¦&𝑧)] = [β– 8(1&0&0@0&1&0@0&0&1)] [β– 8(0(0)+π‘₯(π‘₯)+π‘₯(π‘₯)&0(2𝑦)+π‘₯(𝑦)+π‘₯(βˆ’π‘¦)&0(𝑧)+π‘₯(βˆ’π‘§)+π‘₯(𝑧)@2𝑦(0)+𝑦(π‘₯)βˆ’π‘¦(π‘₯)&2𝑦(2𝑦)+𝑦(𝑦)βˆ’π‘¦(βˆ’π‘¦)&2𝑦(𝑧)+𝑦(βˆ’π‘§)βˆ’π‘¦(𝑧)@𝑧(0)βˆ’π‘§(π‘₯)+𝑧(π‘₯)&𝑧(2𝑦)βˆ’π‘§(𝑦)+𝑧(βˆ’π‘¦)&𝑧(𝑧)βˆ’π‘§(βˆ’π‘§)+𝑧(𝑧))] = [β– 8(1&0&0@0&1&0@0&0&1)] [β– 8(0+π‘₯^2+π‘₯^2&0+π‘₯π‘¦βˆ’π‘₯𝑦&0βˆ’π‘₯𝑧+π‘₯𝑧@0+π‘₯π‘¦βˆ’π‘₯𝑦&4𝑦^2+𝑦^2+𝑦^2&2π‘§π‘¦βˆ’π‘§π‘¦βˆ’π‘§π‘¦@0βˆ’π‘₯𝑧+π‘₯𝑧&2π‘§π‘¦βˆ’π‘§π‘¦βˆ’π‘§π‘¦&𝑧^2+𝑧^2+𝑧^2 )]= [β– 8(1&0&0@0&1&0@0&0&1)] [β– 8(2π‘₯^2&0&0@0&6𝑦^2&0@0&0&3𝑧^2 )]= [β– 8(1&0&0@0&1&0@0&0&1)] Since matrices are equal, corresponding elements are equal Thus, x = Β± 1/√2 , y = Β± 1/√6 , z = Β± 1/√3 2x2 = 1 x2 = 1/2 x = ±√(1/2) x = Β± 𝟏/√𝟐 6y2 = 1 y2 = 1/6 y = ±√(1/6) y = Β± 𝟏/βˆšπŸ” 3z2 = 1 z2 = 1/3 z = ±√(1/3) z = Β± 𝟏/βˆšπŸ‘

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.