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GATE 2019 ME (ME1) – Question 36

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 2 marks · Multiple choice

The equations $x+y+z=1$, $ax-ay+3z=5$, $5x-3y+az=6$ have infinitely many solutions for which a?

  1. −3
  2. 3
  3. 4
  4. −4

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Correct answer: (C) 4

Explanation

At $a=4$, the third equation equals the sum of the first two, including its right-hand side. The first two equations are independent, so $\operatorname{rank}(A)=\operatorname{rank}([A|b])=2<3$. One free variable remains. **$a=4$, C**.