GATE CS 2005 Q49 — Eigenvalues and Eigenvectors, LU Decomposition

We can find the eigenvalues of the matrix $A = \begin{pmatrix} 2 & -1 \\ -4 & 5 \end{pmatrix}$ quickly using the properties of eigenvalues:

1. Sum of eigenvalues (Trace): $$\lambda_1 + \lambda_2 = \text{Trace}(A) = 2 + 5 = 7$$
2. Product of eigenvalues (Determinant): $$\lambda_1 \lambda_2 = \det(A) = (2)(5) - (-1)(-4) = 10 - 4 = 6$$

Now, we check the given options to see which pair of numbers has a sum of $7$ and a product of $6$:

• Option A: $-1$ and $1$ ⇒ Sum = $0$, Product = $-1$ (Incorrect)
• Option B: $1$ and $6$ ⇒ Sum = $7$, Product = $6$ (Correct)
• Option C: $2$ and $5$ ⇒ Sum = $7$, Product = $10$ (Incorrect)
• Option D: $4$ and $-1$ ⇒ Sum = $3$, Product = $-4$ (Incorrect)