R1 → R1 + bR3 (i.e. replace 1st row by addition of 1st row and b times 3rd row)

R2 → R2 - aR3 (i.e. replace 2nd row by subtraction of 2nd row and a times 3rd row)


.


Taking both (1 + a2 + b2) outside the determinant from 1st and 2nd row



Expanding the determinant along 1st row



∴ LHS = (1 + a2 + b2)2 [1 + a2 - b2 - (-2b2)]


∴ LHS = (1 + a2 + b2 )2 (1 + a2 + b2)


∴ LHS = (1 + a2 + b2 )3 = RHS



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