GATE 2022 | MATHS | Q-25

Question

Consider the linear system of equations \(Ax = b\) with
\[ A =
\begin{bmatrix}
3 & 1 & 1 \\
1 & 4 & 1 \\
2 & 0 & 3 \\
\end{bmatrix}
\]
and
\[ b =
\begin{bmatrix}
2 \\
3 \\
4 \\
\end{bmatrix}
\]

Which of the following statements are TRUE?

(A)The Jacobi iterative matrix is
\[ \begin{bmatrix}
0 & \frac{1}{4} & \frac{1}{3} \\
\frac{1}{3} & 0 & \frac{1}{3} \\
\frac{2}{3} & 0 & 0 \\
\end{bmatrix} \]

(B) The Jacobi iterative method converges for any initial vector.

(C) The Gauss-Seidel iterative method converges for any initial vector.

(D) The spectral radius of the Jacobi iterative matrix is less than 1.