The product of all eigenvalues of the matrix \(\begin{bmatrix} 1 & 2 & 3 \\ 4…
2024
The product of all eigenvalues of the matrix \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} \) is
- A.
−1
- B.
0
- C.
1
- D.
2
Attempted by 103 students.
Show answer & explanation
Correct answer: B
Key fact: the product of the eigenvalues of a square matrix equals its determinant.
Compute the determinant: observe the rows (1,2,3), (4,5,6), (7,8,9) are linearly dependent because (7,8,9) = −1×(1,2,3) + 2×(4,5,6).
Linear dependence of rows implies the determinant is 0.
Therefore the product of all eigenvalues (the determinant) is 0.
A video solution is available for this question — log in and enroll to watch it.