Consider the matrix M = bmatrix 2 & -1 \\ 3 & 1 bmatrix. Which ONE of the following statements is TRUE?

The eigenvalues of M are complex conjugate pairs.

The eigenvalues of M are non-negative and real.

The eigenvalues of M are complex conjugate pairs.

One eigenvalue of M is positive and real, and another eigenvalue of M is zero.

One eigenvalue of M is non-negative and real, and another eigenvalue of M is negative and real.

GATE DA 2024 Q13: Nature of eigenvalues of a 2×2 matrix

Consider the matrix $\mathbf{M} = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}$ .

Which ONE of the following statements is TRUE?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: B (official GATE 2024 final answer key)

Concept used: Eigenvalues of a 2×2 matrix from $\lambda^2 - (\operatorname{tr})\lambda + \det = 0$ .

First step: Here trace $= 3$ and determinant $= 2 + 3 = 5$ , so the discriminant is $9 - 20 < 0$ .

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key