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For every real number x with -1 < x < 1, define A(x) = (1 - x)⁻¹ [[1, -x], [-x, 1]]. If z = (x + y)/(1 + xy), then which relation holds?

1. A(z) = A(x) + A(y)
2. A(z) = A(x)[A(y)]⁻¹
3. A(z) = A(x)A(y)
4. A(z) = A(x) - A(y)

Correct answer: A(z) = A(x)A(y)

Solution

Computing A(x)A(y) and simplifying with z=(x+y)/(1+xy) shows A(x)A(y)=A(z). Hence A(z)=A(x)A(y).

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