GATE AE 2022 • Q49 NAT • 2 Marks Engineering Mathematics

Engineering Mathematics

Linear Algebra

Subtopic: Eigenvalues & Eigenvectors

Question Statement:

The largest eigenvalue of the given matrix is _____: $$\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$

Official Answer Key:

2.0

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2022 Q.49)** **1. Eigenvalue Determination**: Characteristic equation $\det(A - \lambda I) = 0$: $$\begin{vmatrix} -\lambda & 1 & 1 \\ 1 & -\lambda & 1 \\ 1 & 1 & -\lambda \end{vmatrix} = -\lambda(\lambda^2 - 1) - 1(-\lambda - 1) + 1(1 + \lambda) = 0$$ $$-(\lambda + 1)^2(\lambda - 2) = 0$$ Eigenvalues are $\lambda = -1, -1, 2$. The largest eigenvalue is $\lambda_{\max} = 2$. - **Official Answer Key**: **2**
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