GATE CS 2015 Set 1 Q36 — Eigenvalues and Eigenvectors, LU Decomposition

We can find the values of $a$ and $b$ using the fundamental properties of eigenvalues of a matrix:

1. Sum of Eigenvalues (Trace): The sum of the eigenvalues of a matrix is equal to the trace (sum of diagonal elements) of the matrix.$$\lambda_1 + \lambda_2 = \text{Tr}(A)$$Given eigenvalues are $\lambda_1 = -1$ and $\lambda_2 = 7$:$$-1 + 7 = 1 + a \implies 6 = 1 + a \implies a = 5$$Checking the options, only Option D has $a = 5$.


2. Product of Eigenvalues (Determinant): The product of the eigenvalues is equal to the determinant of the matrix.$$\lambda_1 \cdot \lambda_2 = \det(A)$$$$(-1) \cdot 7 = (1 \cdot a) - (4 \cdot b)$$Substituting $a = 5$:$$-7 = 5 - 4b \implies 4b = 12 \implies b = 3$$

Thus, $a = 5$ and $b = 3$, which corresponds to Option D.