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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.
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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/sqrt(2),`
`6y^2 = 1, y = ±1/sqrt(6),`
3z2 = 1
∴ `z = ±1/sqrt(3)`
Hence, `x = ±1/sqrt(2), y = ±1/sqrt(6), z = ±1/sqrt(3)`.
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