GATE | GATE CS Mock 2018 | Question 61

The matrix A has (1, 2, 1)^T and (1, 1, 0)^T as eigenvectors, both with eigenvalue 7, and its trace is 2. The determinant of A is __________ .
(A) 84
(B) 588
(C) 49
(D) None of these


Answer: (D)

Explanation: The matrix A is a 3×3 matrix, so it has 3 eigenvalues in total. The eigenspace E7 contains the vectors (1, 2, 1)^T and (1, 1, 0)^T, which are linearly independent. So E7 must have dimension at least 2, which implies that the eigenvalue 7 has multiplicity at least 2.

Let the other eigenvalue be λ, then from the trace λ+7+7 = 2, so λ = −12. So the three eigenvalues are 7, 7 and -12. Hence, the determinant of A is 7 × 7 × −12 = −588.

Option (D) is correct.

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