GATE CS 2024 Set 1 — Question 12
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
The product of all eigenvalues of the matrix is
Correct answer
(B) 0
Solution
The product of all eigenvalues of a square matrix is equal to its determinant. Let . We can calculate the determinant directly:Alternatively, observe that the rows are in an arithmetic progression, which implies linear dependence (). Since the rows are linearly dependent, the determinant is zero. Since the product of eigenvalues equals the determinant, the product is 0.
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