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Question
Find the values of x, y, z if the matrix `A = [(0,2y,z),(x,y,-z),(x , -y,z)]` satisfy the equation A'A = I.
Solution
Here, A = `[(0,2y,z),(x,y,-z),(y,-y,z)]`
⇒ A' = `[(0,x,x),(2y,y,-y),(z,-z,z)]`
∴ A'A = `[(0,x,x),(2y,y,-y),(z,-z,z)][(0,2y,z),(x,y,-z),(x,-y,z)]`
= `[(1,0,0),(0,1,0),(0,0,1)]`
= `[(0 + x^2 + x^2, 0 + xy - xy, -xz + xz),(0 + yz - yx, 4y^2 + y^2 + y^2, 2yz - yz - yz),(0 - zx + 2x, 2yz - zy - zy, z^2 + z^2 + z^2)]`
= `[(1,0,0),(0,1,0),(0,0,1)]`
= `[(2x^2,0,0),(6,6y^2,0),(0,0,3z^2)] = [(1,0,0),(0,1,0),(0,0,1)]`
∴ `2x^2 = 1, x = ±1/sqrt2,`
`6y^2 = 1, y = ±1/sqrt6,`
3z2 = 1
∴ z = ±`1/sqrt3`
Hence, x = `±1/sqrt2, y = ±1/sqrt6, z = ±1/sqrt3`
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