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Part CCSIR NET June 2024diagonal-dominance-is-sufficient-not-necessary

Diagonal dominance is sufficient not necessary

Let S denote the set of all 2 × 2 matrices A such that the iterative sequence generated by the Gauss-Seidel method applied to the system of linear equations converges for every initial guess. Then which of the following statements are true?

  1. A.[[5, 8], [1, 2]] ∈ S✓
  2. B.[[3, 2], [1, 2]] ∈ S✓
  3. C.[[−3, 1], [2, 3]] ∈ S✓
  4. D.[[2, 2], [4, 3]] ∈ S

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Related counterexample: Newton's method converges quadratically to any root

More on this topic

The chapter behind this: Root finding and orders of convergence — free to read

From Numerical Analysis › Root finding: bisection, Newton–Raphson, fixed point, order of convergence

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