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GATE 2022 EC – Question 12

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Consider a system of linear equations $Ax=b$, where

$$A=\begin{bmatrix}1&-\sqrt2&3\\-1&\sqrt2&-3\end{bmatrix},\quad b=\begin{bmatrix}1\\3\end{bmatrix}.$$

This system of equations admits ________.

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

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Correct answer: (C) no solutions for $x$

Explanation

Adding the two equations gives $0=1+3=4$, which is impossible. Equivalently, rank of $A$ is 1 while rank of the augmented matrix $[A|b]$ is 2, so the system is inconsistent and has no solution.