If the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is TRUE? A. p - q + r = 0 B. p + q - r = 0 C. p + q + r = 0 or p = q = r D. p - q + r = 0 and p = q = r

GATE 2015 · Engineering Mathematics · System of Equations · medium

Answer: The homogeneous system has non-trivial solutions iff p+q+r=0 or p=q=r. Answer: C.

1. Write the coefficient matrix: The system matrix is A = [[p,q,r],[q,r,p],[r,p,q]]. This is a circulant matrix.
2. Compute det(A) using cofactor expansion: det(A) = p(rq-p^2) - q(q^2-rp) + r(qp-r^2) = prq - p^3 - q^3 + qrp + rqp - r^3 = 3pqr - p^3 - q^3 - r^3
3. Factor the determinant expression: det(A) = -(p+q+r)(p^2+q^2+r^2-pq-qr-rp). For non-trivial solution: det(A)=0. So either (p+q+r)=0 or (p^2+q^2+r^2-pq-qr-rp)=0.
4. State the conditions: Non-trivial solution exists iff p+q+r=0 or p=q=r. This matches option C.