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Using the property of determinants and without expanding, prove that |(1 + a^2 - b^2, 2ab, -2b),(2ab, 1 - a^2 + b^2, 2a), (2b, -2a, 1 - a^2 - b^2)|

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Using the property of determinants and without expanding, prove that 

|(1 + a2 - b2, 2ab, -2b),(2ab, 1 - a2 + b2, 2a), (2b, -2a, 1 - a2 - b2)| = (1 + a2 + b2)3

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(Using C1 → C1 - bC3 and C2 → C2 + aC3)

(R3 → R3 - bR1 + aR2)

Expanding along R1, we get

= (1 + a2 + b2)2[1(1 + a2 + b2)]

= (1 + a2 + b2)3 = RHS

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