The GATE Grind

GATE 2023 IN – Question 11

Engineering Mathematics · Linear algebra and matrices · 1 mark · Multiple choice

Choose the solution set $S$ of $x-2y+z=0$ and $x-z=0$. Here $\mathbb R$ denotes the real numbers; tuples denote column vectors.

  1. $S=\{\alpha(1,1,1)^T:\alpha\in\mathbb R\}$
  2. $S=\{\alpha(1,1,1)^T+\beta(1,0,1)^T:\alpha,\beta\in\mathbb R\}$
  3. $S=\{\alpha(1,1,1)^T+\beta(2,1,2)^T:\alpha,\beta\in\mathbb R\}$
  4. $S=\{\alpha(1,0,1)^T:\alpha\in\mathbb R\}$

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Show answer and explanation

Correct answer: (A) $S=\{\alpha(1,1,1)^T:\alpha\in\mathbb R\}$

Explanation

The second equation gives $z=x$. Substitution into the first yields $2x-2y=0$, hence $y=x$. All solutions are $\alpha(1,1,1)^T$, $\alpha\in\mathbb R$. Answer: **A**.