Discussion
Let
\[ A = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 2 & -1 \\ -1 & 1 & 2 \end{pmatrix} \]
and \(b\) be a \(3 \times 1\) real column vector. Consider the statements.
I. The Jacobi iteration method for the system \((A + \varepsilon I_{3})x = b\) converges for any initial approximation and \(\varepsilon \gt 0\).
II. The Gauss-Seidel iteration method for the system \((A + \varepsilon I_{3})x = b\) converges for any initial approximation and \(\varepsilon \gt 0\).
Which one of the following is correct?
Answer: A
Both I and II are TRUE.
In \(A + \varepsilon I_{3}\) each diagonal entry is \(2 + \varepsilon\), and the off-diagonal entries in each row have absolute values summing to \(1 + 1 = 2\). Since \(2 + \varepsilon \gt 2\) for \(\varepsilon \gt 0\), the matrix is strictly diagonally dominant by rows.
For a strictly diagonally dominant matrix both the Jacobi and the Gauss-Seidel iterations converge for every starting vector. So I and II both hold, and the options denying either are wrong.
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