GATE 2025 MA – Question 36
For Ax=b with nonzero diagonal D, $D^{-1}A=\begin{pmatrix}1&1&-2\\3&1&2\\1&1&1\end{pmatrix}$ and D⁻¹b=(4,4,1)ᵀ. S1: Jacobi from x⁽⁰⁾=(1,1,1)ᵀ gives x⁽¹⁾=(5,−1,−1)ᵀ. S2: Jacobi diverges for some initial vector. Choose the truth values.
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Correct answer: (C) Both true
Explanation
Jacobi matrix is $B=I-D^{-1}A=\begin{pmatrix}0&-1&2\\-3&0&-2\\-1&-1&0\end{pmatrix}$. B(1,1,1)ᵀ+(4,4,1)ᵀ=(5,−1,−1)ᵀ, proving S1. Its characteristic polynomial λ³−3λ−4 is negative at 2 and positive at 3, so B has a real eigenvalue >2. An initial error along its eigenvector grows, proving S2. **C**.