GATE CS 2015 Set 3 Q.43 (CS Q.33): Linear Algebra

If the following system has non-trivial solution,

\begin{aligned} p x + q y + r z & = 0 \\ q x + r y + p z & = 0 \\ r x + p y + q z & = 0 \end{aligned}

then which one of the following options is TRUE?

$p - q + r = 0$ or $p = q = -r$

$p + q - r = 0$ or $p = -q = r$

$p + q + r = 0$ or $p = q = r$

$p - q + r = 0$ or $p = -q = -r$

Official answer and solution

Solution

Set the determinant of the coefficient matrix to zero and factorize the resulting cubic polynomial.

A homogeneous linear system $A X = 0$ has a non-trivial solution if and only if $\det(A) = 0$ :

\det \begin{bmatrix} p & q & r \\ q & r & p \\ r & p & q \end{bmatrix} = 0

Compute the determinant of this circulant matrix:

\begin{aligned} \det(A) & = p(r q - p^2) - q(q^2 - p r) + r(q p - r^2) \\ & = p q r - p^3 - q^3 + p q r + p q r - r^3 \\ & = 3 p q r - (p^3 + q^3 + r^3) \\ & = -(p^3 + q^3 + r^3 - 3 p q r) \end{aligned}

Using the algebraic factorization identity:

p^3 + q^3 + r^3 - 3 p q r = (p + q + r)(p^2 + q^2 + r^2 - p q - q r - r p)

= \frac{1}{2}(p + q + r)\left((p - q)^2 + (q - r)^2 + (r - p)^2\right)

For the determinant to be zero, either:

Thus, the condition is $p + q + r = 0$ or $p = q = r$ (Option C).

Final answer

Common trap

Sign error in expanding the 3x3 determinant or forgetting that the sum of three non-negative squares is zero only when each term is zero (p = q = r).

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