Question

Which of the following statements is/are true?
(I) Gauss-Seidel method for solving a linear system converges faster than the Gauss-Jacobi method.
(II) In the Gauss-Jordan method for solving a linear system, the coefficient matrix is transformed into an upper triangular matrix.

• A. Only (I)
• B. Only (II)
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

Recall that Gauss-Jordan eliminates entries above the pivot too, going all the way to a diagonal matrix.

Answer 1

The Correct Option is A

Step 1: Check statement (I). The Gauss-Jacobi method computes every component of a new iterate using only values from the previous iteration, while the Gauss-Seidel method uses the most recently updated components as soon as they are available within the same iteration. Because of this immediate use of updated information, Gauss-Seidel generally converges faster than Gauss-Jacobi for systems where both methods converge, such as diagonally dominant systems. So statement (I) is true.

Step 2: Check statement (II). The Gauss-Jordan method eliminates the unknowns both below and above each pivot, reducing the coefficient matrix all the way to a diagonal matrix (the identity matrix after normalisation), so the solution can be read off directly without back substitution. Reducing a matrix only to upper triangular form, followed by back substitution, is what plain Gaussian elimination does, not Gauss-Jordan elimination. So statement (II) is false.

Step 3: Since (I) is true and (II) is false, only statement (I) holds.

\[\boxed{\text{Only (I)}}\]