GATE 2024 DA Q.13: solution and answer

GATE 2024 · DAQ.13 · 1 mark · MCQ

Consider the matrix $M = [23 - 1 1]$ .

Which ONE of the following statements is TRUE?

(A)
The eigenvalues of $M$ are non-negative and real.
(B)
The eigenvalues of $M$ are complex conjugate pairs.
(C)
One eigenvalue of $M$ is positive and real, and another eigenvalue of $M$ is zero.
(D)
One eigenvalue of $M$ is non-negative and real, and another eigenvalue of $M$ is negative and real.

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Answer and solution

ANSWER

The eigenvalues of $M$ are $\frac{3 \pm i 11}{2}$ , a complex conjugate pair, so option (B) is correct.

Official answer: BSource: Official answer key, GATE 2024 DA (IISc Bengaluru)

For a $2 \times 2$ matrix the characteristic equation is $λ^{2} - (tr M) λ + det M = 0$ .
Trace: $tr M = 2 + 1 = 3$ .
Determinant: $det M = (2) (1) - (- 1) (3) = 2 + 3 = 5$ .
So the characteristic equation is $λ^{2} - 3 λ + 5 = 0$ .
Discriminant: $D = 3^{2} - 4 (5) = 9 - 20 = - 11 < 0$ , so there are no real roots.
$λ = \frac{3 \pm i 11}{2}$ , a complex conjugate pair. Check: the sum is $3 = tr M$ and the product is $\frac{9 + 11}{4} = 5 = det M$ .
Options (A), (C) and (D) each require at least one real eigenvalue, so they are false. In (C), $det M = 5 \neq = 0$ also rules out a zero eigenvalue. Only (B) is true.

KEY IDEA

For a $2 \times 2$ real matrix, find the sign of the discriminant of $λ^{2} - tr (M) λ + det (M)$ . A negative discriminant means a complex conjugate pair, with no real eigenvalue.

WATCH OUT FOR

Computing the determinant as $2 - 3 = - 1$ by dropping the sign of the $- 1$ entry. That gives $λ^{2} - 3 λ - 1 = 0$ , which has real roots of opposite sign and wrongly points to (D). The correct determinant is $(2) (1) - (- 1) (3) = 5$ .

From the GATE 2024 DA paper. Every question of this paper · Download the full paper · All 6 Eigenvalues questions, with notes

GATE 2024 DA Q.13: solution and answer

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