The GATE Grind

GATE 2014 EC (EC2) – Question 36

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

The linear system $\begin{pmatrix}2&1&3\\3&0&1\\1&2&5\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}=\begin{pmatrix}5\\-4\\14\end{pmatrix}$ has ______.

  1. a unique solution
  2. infinitely many solutions
  3. no solution
  4. exactly two solutions

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Show answer and explanation

Correct answer: (B) infinitely many solutions

Explanation

The third coefficient row is twice the first minus the second:(1,2,5)=2(2,1,3)−(3,0,1). The right-hand side obeys the same relation14=2×5−(−4), so the system is consistent but has rank2in3unknowns. Therefore it has **infinitely many solutions,B**.