The GATE Grind

GATE 2024 IN – Question 19

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

A matrix $M=[v_1\ v_2\ v_3]$ is formed by stacking three column vectors. Which set gives $\operatorname{rank}(M)=3$? Tuples below denote column vectors.

  1. $v_1=(1,0,1)^T,\ v_2=(0,-1,0)^T,\ v_3=(1,-1,1)^T$
  2. $v_1=(1,1,1)^T,\ v_2=(-1,0,1)^T,\ v_3=(0,0,0)^T$
  3. $v_1=(1,0,1)^T,\ v_2=(-1,0,1)^T,\ v_3=(1,-1,1)^T$
  4. $v_1=(1,1,1)^T,\ v_2=(-1,1,-1)^T,\ v_3=(0,-1,0)^T$

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Correct answer: (C) $v_1=(1,0,1)^T,\ v_2=(-1,0,1)^T,\ v_3=(1,-1,1)^T$

Explanation

In C, $v_1=(1,0,1)^T$ and $v_2=(-1,0,1)^T$ are independent in the xz-plane, while $v_3=(1,-1,1)^T$ has a nonzero y-component.

Thus all three are independent. In A, $v_3=v_1+v_2$; B has a zero column; D has identical first and third rows.

Answer: **C**.