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Question
Find x, y, z if A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfies A′ = A–1.
Solution
Matrix A is such that A′ = A–1
⇒ AA' = I
⇒ `[(0, 2y, z),(x, y, -z),(x, -y, z)] [(0, x, x),(2y, y, -y),(z, -z, z)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
⇒ `[(4y^2 + z^2, 2y^2 - z^2, -2y^2 + z^2),(2y^2 - z^2, x^2 + y^2 +z^2, x^2 - y^2 - z^2),(-2^2 + z^2, x^2 - y^2 + z^2, x^2 + y^2 + z^2)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
⇒ `4y^2 + z^2` = 1
`2y^2 - z^2` = 0
`x^2 + y^2 + z^2` = 1
`x^2 - y^2 - z^2`= 0
⇒ `y^2 = 1/6, z^2 = 1/3, x^2 = 1/2`
⇒ x = `+- 1/sqrt(2)`
⇒ y = `+- 1/sqrt(6)`
And z = `+- 1/sqrt(3)`
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