GATE CS 2015 Set 3 Q33 — Linear Algebra: Matrices, Determinants, System of Linear Equations

For a homogeneous system of linear equations to have a non-trivial solution, the determinant of its coefficient matrix must be equal to zero.

The given system of equations is:

• $px + qy + rz = 0$
• $qx + ry + pz = 0$
• $rx + py + qz = 0$

The coefficient matrix $A$ is:

Setting the determinant of $A$ to zero:

Expanding the determinant:

Using the standard algebraic factorization identity:

This gives two possible cases:

1. Case 1: $$p + q + r = 0$$
2. Case 2: $$p^2 + q^2 + r^2 - pq - qr - rp = 0$$ Multiplying by $2$, we can rewrite this as: $$(p-q)^2 + (q-r)^2 + (r-p)^2 = 0$$ Since the sum of squares of real numbers is zero if and only if each term is individually zero, we have: $$p - q = 0, \quad q - r = 0, \quad r - p = 0 \implies p = q = r$$

Thus, the system has a non-trivial solution if and only if:

Therefore, Option C is the correct answer.