Find the product of the eigenvalues of pmatrix0&0&1\\0&1&0\\1&0&0pmatrix.

GATE 2020 PH – Question 36

Mathematical Physics · Matrices: similarity transformations, diagonalization, eigenvalues and eigen vectors · 2 marks · Multiple choice

Find the product of the eigenvalues of $$\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}.$$

−1
1
0
2

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Correct answer: (A) −1

Explanation

The product of eigenvalues equals the determinant. This matrix exchanges the first and third basis vectors while leaving the second fixed. One interchange has determinant −1. Equivalently, its eigenvalues are 1, 1 and −1. Their product is −1, option A.

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