GATE CS 2010 Q29 — Eigenvalues and Eigenvectors, LU Decomposition

We can solve this problem efficiently using the fundamental properties of eigenvalues:

1. Sum of Eigenvalues (Trace):

   The sum of the eigenvalues of a matrix is equal to the trace (the sum of the diagonal elements) of the matrix.

   $$\lambda_1 + \lambda_2 = a_{11} + a_{22}$$ $$4 + 8 = 2 + y \implies 12 = 2 + y \implies y = 10$$
2. Product of Eigenvalues (Determinant):

   The product of the eigenvalues of a matrix is equal to the determinant of the matrix.

   $$\lambda_1 \cdot \lambda_2 = \det(A)$$ $$4 \cdot 8 = (2 \cdot y) - (3 \cdot x)$$

   Substitute $y = 10$ into the equation:

   $$32 = 2(10) - 3x$$ $$32 = 20 - 3x$$ $$3x = 20 - 32 = -12 \implies x = -4$$

Thus, the values are $x = -4$ and $y = 10$, which corresponds to Option D.