GATE CS 2007 Q25 — Eigenvalues and Eigenvectors, LU Decomposition

For a block matrix of the form $M = \begin{bmatrix} A & B \\ B & A \end{bmatrix}$ where the submatrices $A$ and $B$ commute (i.e., $AB = BA$), the eigenvalues of $M$ are given by the union of the eigenvalues of $(A + B)$ and $(A - B)$.

In this problem, $B = I$. Since the identity matrix $I$ commutes with any matrix $A$, we can apply this property directly:

• The eigenvalues of $A + I$ are of the form $\lambda + 1$, where $\lambda$ is an eigenvalue of $A$.
• The eigenvalues of $A - I$ are of the form $\lambda - 1$, where $\lambda$ is an eigenvalue of $A$.

Given the eigenvalues of $A$ are $\{-5, -2, 1, 4\}$:

• Eigenvalues of $A + I$: $-5+1 = -4$,\quad $-2+1 = -1$,\quad $1+1 = 2$,\quad $4+1 = 5$
• Eigenvalues of $A - I$: $-5-1 = -6$,\quad $-2-1 = -3$,\quad $1-1 = 0$,\quad $4-1 = 3$

Thus, the complete set of eigenvalues for the block matrix is $\{-6, -4, -3, -1, 0, 2, 3, 5\}$. Among the given options, only 2 (Option C) is an eigenvalue of the block matrix.