GATE CS 2024 Set 1 | Question 12

MCQ · 1 mark · −0.33

The product of all eigenvalues of the matrix \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\) is

  1. Option A: \(-1\)
  2. Option B: 0 (correct)
  3. Option C: 1
  4. Option D: 2

Answer

(B)

Explanation

The product of all eigenvalues of a square matrix equals its determinant.

Here the rows are linearly dependent: each row exceeds the previous one by \((3, 3, 3)\), so \(R_{1} + R_{3} = 2R_{2}\). A matrix with dependent rows has determinant 0:

\[\det\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} = 0.\]

So 0 is one of the eigenvalues, and the product of all the eigenvalues is 0, whatever the other two are.

Answer: B