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GATE 2025 EE – Question 13

Engineering Mathematics · Linear Algebra: Systems of linear equations · 1 mark · Multiple choice

Let $A=\begin{bmatrix}1&1&1\\-1&-1&-1\\0&1&-1\end{bmatrix}$, and $\boldsymbol b=\begin{bmatrix}1/3\\-1/3\\0\end{bmatrix}$. Then, the system of linear equations $A\boldsymbol x=\boldsymbol b$ has

  1. a unique solution.
  2. infinitely many solutions.
  3. a finite number of solutions.
  4. no solution.

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Show answer and explanation

Correct answer: (B) infinitely many solutions.

Explanation

Row 2 of $A$ is $-1\times$ row 1 and $b_2=-b_1$, so the second equation is redundant. The system reduces to $x+y+z=\tfrac13$ and $y-z=0$: two independent equations in three unknowns, and it is consistent. So there are infinitely many solutions.