Question

Let \( A \) be a \( 3 \times 3 \) real matrix and \( b \) be a \( 3 \times 1 \) real column vector. Consider the statements:
1. The Jacobi iteration method for the system \( (A + \epsilon I_3)x = b \) converges for any initial approximation and \( \epsilon>0 \).
2. The Gauss-Seidel iteration method for the system \( (A + \epsilon I_3)x = b \) converges for any initial approximation and \( \epsilon>0 \).
Which one of the following is correct?

• A. Both I and II are TRUE
• B. I is TRUE and II is FALSE
• C. I is FALSE and II is TRUE
• D. Both I and II are FALSE

Hint:

For iterative methods, check conditions like diagonal dominance or positive definiteness to ensure convergence.

Answer 1

The Correct Option is A

Step 1: Understanding the iteration methods. The Jacobi and Gauss-Seidel methods for solving \( Ax = b \) converge if \( A \) is strictly diagonally dominant or positive definite. Adding \( \epsilon I_3 \) ensures that \( A + \epsilon I_3 \) becomes strictly diagonally dominant for \( \epsilon>0 \). Step 2: Verifying convergence conditions. - For Jacobi method: \( (A + \epsilon I_3) \) is strictly diagonally dominant, ensuring convergence. - For Gauss-Seidel method: \( (A + \epsilon I_3) \) being strictly diagonally dominant also ensures convergence. Step 3: Conclusion. Both statements are true. The correct answer is \( {(1)} \).