GATE CS 2024 Set 1 Q2 — Eigenvalues and Eigenvectors, LU Decomposition

We are given the matrix:

A fundamental property of eigenvalues states that the product of all eigenvalues of a matrix is equal to its determinant:

Let's perform row operations on $A$ to simplify the determinant calculation:

1. Perform $R_2 \leftarrow R_2 - R_1$: $$\begin{bmatrix} 1 & 2 & 3 \\ 3 & 3 & 3 \\ 7 & 8 & 9 \end{bmatrix}$$
2. Perform $R_3 \leftarrow R_3 - R_2$ (using the original $R_2$): $$\begin{bmatrix} 1 & 2 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{bmatrix}$$

Since the second and third rows of the modified matrix are identical, the rows of the matrix are linearly dependent. This implies that the matrix is singular, and its determinant is zero:

Therefore, the product of all eigenvalues of the matrix is $0$.