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GATE 2022 MA – Question 25

Numerical Analysis · Linear Systems · 1 mark · Multiple select

For Ax=b, $A=\begin{pmatrix}3&1&1\\1&4&1\\2&0&3\end{pmatrix}$ and b=(2,3,4)ᵀ, select all true statements.

  1. The Jacobi matrix is $\begin{pmatrix}0&1/4&1/3\\1/3&0&1/3\\2/3&0&0\end{pmatrix}$.
  2. Jacobi converges from every initial vector.
  3. Gauss–Seidel converges from every initial vector.
  4. The Jacobi spectral radius is below 1.

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Correct answer: (B) Jacobi converges from every initial vector.; (C) Gauss–Seidel converges from every initial vector.; (D) The Jacobi spectral radius is below 1.

Explanation

A is strictly row diagonally dominant, so both iterative methods converge from every start. The actual Jacobi matrix is $\begin{pmatrix}0&-1/3&-1/3\\-1/4&0&-1/4\\-2/3&0&0\end{pmatrix}$, refuting A. Its infinity norm is 2/3, bounding its spectral radius below 1. **B,C,D**.

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