GATE 2015 CS (CS3) – Question 52
The homogeneous equations px+qy+rz=0, qx+ry+pz=0, rx+py+qz=0 have a nonzero real solution. Which condition follows for real p,q,r?
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Correct answer: (C) p+q+r=0 or p=q=r
Explanation
A nonzero solution requires zero determinant. Up to sign it factors as (p+q+r)(p²+q²+r²−pq−qr−rp). The second factor equals half of (p−q)²+(q−r)²+(r−p)², so over the reals it vanishes exactly when p=q=r. Hence C.