GATE CS 2011 Q40 — Eigenvalues and Eigenvectors, LU Decomposition

The given matrix is:

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 0 & 0 & 3 \end{bmatrix}$$

An important property of matrices is that for any triangular matrix (either upper triangular or lower triangular), the eigenvalues are simply the elements on its main diagonal.

Looking at the given matrix $A$:

• All elements below the main diagonal are zero ($a_{21} = 0$, $a_{31} = 0$, $a_{32} = 0$).
• Therefore, $A$ is an upper triangular matrix.

The diagonal elements of $A$ are $1$, $4$, and $3$.

Thus, the eigenvalues of the matrix are $1, 4, 3$.

Hence, the correct option is A.