GATE 2025 ST – Question 56

Matrix Theory · Eigenvalues and Matrix Decompositions · 2 marks · Numerical answer

A 5×5 P has zeros on the diagonal, ones above and minus ones below. Find the product of its five eigenvalues (integer).

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Correct answer: 0 to 0

Explanation

P is skew-symmetric of odd order: Pᵀ=−P. Therefore det P=det Pᵀ=det(−P)=−det P, forcing det P=0. The eigenvalue product equals the determinant, hence **0**.

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