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GATE 2024 DA – Question 13

Linear Algebra · Eigenvalues, eigenvectors and determinant · 1 mark · Multiple choice

Consider the matrix $\boldsymbol{M} = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}$. Which ONE of the following statements is TRUE?

  1. The eigenvalues of M are non-negative and real.
  2. The eigenvalues of M are complex conjugate pairs.
  3. One eigenvalue of M is positive and real, and another eigenvalue of M is zero.
  4. One eigenvalue of M is non-negative and real, and another eigenvalue of M is negative and real.

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Correct answer: (B) The eigenvalues of M are complex conjugate pairs.

Explanation

The trace is 3 and the determinant is $2 \cdot 1 - (-1)(3) = 5$. The characteristic equation $\lambda^2 - 3\lambda + 5 = 0$ has the discriminant $9 - 20 = -11 < 0$, so the eigenvalues are a complex conjugate pair, $\frac{3 \pm i\sqrt{11}}{2}$.