GATE CS 2015 Set 2 Q.15 (CS Q.5): Linear Algebra

The larger of the two eigenvalues of the matrix $\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ is _____.

Your answer

Official answer and solution

Solution

Compute the eigenvalues by solving the characteristic equation $\det(A - \lambda I) = 0$ .

For the matrix $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ , the characteristic equation is:

\det(A - \lambda I) = \det\begin{bmatrix} 4 - \lambda & 5 \\ 2 & 1 - \lambda \end{bmatrix} = 0.

Expanding the determinant:

(4 - \lambda)(1 - \lambda) - (5)(2) = 0

\lambda^2 - 5\lambda + 4 - 10 = 0

\lambda^2 - 5\lambda - 6 = 0.

Factoring the quadratic:

(\lambda - 6)(\lambda + 1) = 0 \implies \lambda_1 = 6,\; \lambda_2 = -1.

The larger eigenvalue is $\max(6, -1) = 6$ .

Final answer

Common trap

Selecting the sum of eigenvalues ( $4+1=5$ ) or forgetting to subtract the determinant product.

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