Consider the system of equations $AX = B$, where $$A = \begin{bmatrix}2 & c & 0 \\ c & 1 & c \\ 0 & c & 2\end{bmatrix} $$ and $c ∈ \Bbb R \setminus \{−2, −1, 1, 2\}$. Which of the following statement is true?
The answer is: Jacobi's method generates a convergent succession for $X$ if and only if $c ∈] - 1, 1 [$.
My doubts is: I've done it and got $$ G = \begin{bmatrix} 0 &-1/2c& 0 \\ -c & 0 & -c\\ 0 &-1/2c & 0 \end{bmatrix}$$ where $G$ is a 3×3 matrix
and so $\|G\|_\infty= \max\bigl(|1/2*c|,|2c|,1/|2|c\bigr) = 2|c| < 1 \iff |c| < 1/2$
why is this wrong?
The statements are:
a) Jacobi's method generates a convergent succession for X if and only if c ∈] −1/√2 , 1/√2 [
b) Jacobi's method generates a convergent succession for X if and only if c ∈] −1/2,1/2 [
c) Jacobi's method generates a convergent succession for X if and only if c ∈] - 1, 1 [;
d) Jacobi's method generates a convergent succession for X if and only if c ∈] - 2, 2 [;