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GATE 2024 ME – Question 36

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

The matrix $\begin{bmatrix} 1 & a \\ 8 & 3 \end{bmatrix}$ (where $a > 0$) has a negative eigenvalue if $a$ is greater than

  1. $\frac{3}{8}$
  2. $\frac{1}{8}$
  3. $\frac{1}{4}$
  4. $\frac{1}{5}$

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Correct answer: (A) $\frac{3}{8}$

Explanation

The trace is $1 + 3 = 4 > 0$, so the two eigenvalues cannot both be negative. One of them is negative exactly when the product of the eigenvalues, which is the determinant, is negative: $3 - 8a < 0$, so $a > \frac{3}{8}$.