GATE DA 2024 | Question 13

MCQ · 1 mark · −0.33

Consider the matrix \(\boldsymbol{M} = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}\).

Which ONE of the following statements is TRUE?

  1. Option A: The eigenvalues of \(\boldsymbol{M}\) are non-negative and real.
  2. Option B: The eigenvalues of \(\boldsymbol{M}\) are complex conjugate pairs. (correct)
  3. Option C: One eigenvalue of \(\boldsymbol{M}\) is positive and real, and another eigenvalue of \(\boldsymbol{M}\) is zero.
  4. Option D: One eigenvalue of \(\boldsymbol{M}\) is non-negative and real, and another eigenvalue of \(\boldsymbol{M}\) is negative and real.

Answer

(B)

Explanation

For a \(2 \times 2\) matrix the characteristic equation is \(\lambda^{2} - (\operatorname{tr}\boldsymbol{M})\lambda + \det \boldsymbol{M} = 0\). Here \(\operatorname{tr}\boldsymbol{M} = 2 + 1 = 3\) and \(\det \boldsymbol{M} = 2 \cdot 1 - (-1) \cdot 3 = 5\), so

\[\lambda^{2} - 3\lambda + 5 = 0, \qquad \lambda = \frac{3 \pm \sqrt{9 - 20}}{2} = \frac{3 \pm i\sqrt{11}}{2}.\]

The discriminant is negative, so the eigenvalues form a complex conjugate pair and none is real, which rules out (A), (C) and (D). A zero eigenvalue would also need \(\det \boldsymbol{M} = 0\).

Answer: B