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GATE 2022 AE – Question 49

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 2 marks · Numerical answer

The largest eigenvalue of $$\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}$$ is ________.

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Correct answer: 2

Explanation

The matrix $A=\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}$ equals $J-I$, where $J$ is the all-ones matrix.

**Eigenvalues.**
- $J$ has eigenvalues 3 (eigenvector $(1,1,1)^T$) and 0 (twice).
- So $A=J-I$ has eigenvalues $3-1=2$ and $0-1=-1$ (twice).

**Check with the trace.** The trace of $A$ is 0 and $2+(-1)+(-1)=0$. ✓
**Check with the eigenvector.** $A(1,1,1)^T=(2,2,2)^T=2(1,1,1)^T$. ✓

The largest eigenvalue is **2**.