GATE 2026 EE Q.30: solution and answer

GATE 2026 · EEQ.30 · 1 mark · MSQ

Two $n \times n$ matrices $A$ and $B$ have a common eigenvalue $2$ , and the same corresponding nonzero eigenvector.

Which of the following options is/are correct?

(Note: $I$ is the $n \times n$ identity matrix.)

(A)
Determinant $(A - 2 I) = 0$
(B)
Determinant $(B - 2 I) = 0$
(C)
Determinant $(A + B - 2 I) = 0$
(D)
Determinant $(A + B - 4 I) = 0$

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Answer and solution

ANSWER

Options (A), (B) and (D) are correct: $det (A - 2 I) = 0$ , $det (B - 2 I) = 0$ and $det (A + B - 4 I) = 0$ .

Official answer: A, B, DSource: Official answer key, GATE 2026 EE (IIT Guwahati)

Let $x \neq = 0$ be the common eigenvector. By the definition of an eigenpair, $A x = 2 x$ and $B x = 2 x$ .
Option (A): $A x = 2 x$ gives $(A - 2 I) x = 0$ with $x \neq = 0$ . So $A - 2 I$ has a nontrivial null space, which means it is singular, and $det (A - 2 I) = 0$ . Correct.
Option (B): the same argument with $B x = 2 x$ gives $(B - 2 I) x = 0$ with $x \neq = 0$ , so $det (B - 2 I) = 0$ . Correct.
Because the eigenvector is the same for both matrices, the two eigen-equations can be added: $(A + B) x = A x + B x = 2 x + 2 x = 4 x$ So $x$ is an eigenvector of $A + B$ with eigenvalue $4$ .
Option (D): $(A + B - 4 I) x = 4 x - 4 x = 0$ with $x \neq = 0$ , so $A + B - 4 I$ is singular and $det (A + B - 4 I) = 0$ . Correct.
Option (C): $(A + B - 2 I) x = 4 x - 2 x = 2 x \neq = 0$ , so the shared eigenvector tells us nothing about whether $2$ is an eigenvalue of $A + B$ . Counter-example: $A = B = 2 I$ satisfies the given data, but $A + B - 2 I = 2 I$ and $det (2 I) = 2^{n} \neq = 0$ . So (C) is not necessarily true.
Correct options: (A), (B) and (D).

KEY IDEA

$λ$ is an eigenvalue of $M$ exactly when $det (M - λ I) = 0$ . If $A$ and $B$ share the eigenvector $x$ with eigenvalues $λ$ and $μ$ , then $(A + B) x = (λ + μ) x$ , so $A + B$ has eigenvalue $λ + μ$ (here $2 + 2 = 4$ ). Eigenvalues add only because the eigenvector is common.

WATCH OUT FOR

Ticking (C) because $2$ is an eigenvalue of both $A$ and $B$ , as if it must then be an eigenvalue of $A + B$ too. The shared eigenvector gives $A + B$ the eigenvalue $4$ , not $2$ . $det (A + B - 2 I)$ can be zero for some particular matrices (for example $A = B = [2001]$ gives $A + B = [4002]$ ), but it fails for $A = B = 2 I$ , so (C) is not guaranteed. A second trap is assuming eigenvalues of $A + B$ are always sums of eigenvalues of $A$ and $B$ . That holds only for a shared eigenvector, which is given here.

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GATE 2026 EE Q.30: solution and answer

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