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GATE 2022 ME (ME1) – Question 46

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

The system of linear equations in real $(x, y)$ given by $(x\ \ y)\begin{bmatrix} 2 & 5 - 2\alpha \\ \alpha & 1 \end{bmatrix} = (0\ \ 0)$ involves a real parameter $\alpha$ and has infinitely many non-trivial solutions for special value(s) of $\alpha$. Which one or more among the following options is/are non-trivial solution(s) of $(x, y)$ for such special value(s) of $\alpha$?

  1. $x = 2$, $y = -2$
  2. $x = -1$, $y = 4$
  3. $x = 1$, $y = 1$
  4. $x = 4$, $y = -2$

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Correct answer: (A) $x = 2$, $y = -2$; (B) $x = -1$, $y = 4$

Explanation

A non-trivial solution exists when the determinant is zero: $2 - \alpha(5 - 2\alpha) = 0$, that is $2\alpha^2 - 5\alpha + 2 = 0$, so $\alpha = 2$ or $\alpha = 0.5$. The equations are $2x + \alpha y = 0$ and $(5 - 2\alpha)x + y = 0$. For $\alpha = 2$: $y = -x$, which gives $(2, -2)$. For $\alpha = 0.5$: $y = -4x$, which gives $(-1, 4)$. The pairs (1, 1) and (4, −2) satisfy neither.