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Part CCSIR NET December 2025gauss-seidel-for-a-2x2-system-converges-exactly-when-the-off-diagonal-product-is-smaller-than-the-diagonal-product

Gauss seidel for a 2x2 system converges exactly when the off diagonal product is smaller than the diagonal product

Let S be the set of all 2×2 matrices A such that the iterative sequence generated by the Gauss-Seidel method converges for every initial guess, when employed to solve the system of equations A(x1,x2)T=(1,2)TA(x_{1},x_{2})^{T}=(1,2)^{T}. Then which of the following statements are true?

  1. A.(4,1;2,3) ∈ S✓
  2. B.(1,2;3,4) ∈ S
  3. C.(1,5;1,10) ∈ S✓
  4. D.(−5,2;1,−4) ∈ 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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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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