Which of the following statements is true regarding Naïve Gaussian elimination?

Question

BlackTom AI · Accepted Answer

Gaussian elimination transforms a linear system into an upper-triangular system by forward elimination (using pivot elements) and then recovers unknowns by back substitution. The "naïve" form performs these steps without any row pivoting or pivot selection rules.

1) Relevant knowledge points and concepts
- Forward elimination: zero out subdiagonal entries using multipliers m_ij = a_ij / a_kk where a_kk is the pivot.
- Back substitution: solve the triangular system from bottom to top.
- Pivoting (partial or full): swapping rows/columns to avoid zero or very small pivots and improve numerical stability.
- "Naïve Gaussian elimination" means no pivoting is done.

2) Step-by-step reasoning
- During forward elimination, the algorithm must divide by the diagonal element a_11, then a_22 (after elimination), etc.
- If any pivot a_kk = 0, computing multiplier m_ik = a_ik / a_kk requires division by zero and the algorithm fails.
- Because naive elimination does not swap rows to avoid zero pivots, it can indeed encounter a zero pivot.

3) Example with intermediate steps
Consider the 2×2 system:
[0  1] [x] = [1]
[1  2] [y]   [3]
Here a_11 = 0. The first elimination multiplier would be m_21 = a_21 / a_11 = 1 / 0, which is undefined (division by zero). A safe procedure would swap rows (pivoting) so the pivot is nonzero, but naïve elimination does not do this.

4) Why the correct answer is correct
- Statement (a) "It can lead to division by zero" is true because naive Gaussian elimination uses the current diagonal entry as pivot without checking or swapping; if that entry is zero, division by zero occurs.

5) Why other options are incorrect
- (b) False: Naïve Gaussian uses back substitution after forward elimination.
- (c) False: This is a joke; it was developed historically but not by Gauss’s younger sibling.
- (d) False: Naïve Gaussian explicitly performs forward elimination.
- (e) False: Naïve Gaussian does not use pivoting; pivoting is the protection against zero pivots.

6) Conclusion
Because naïve Gaussian elimination uses diagonal pivots without row swaps, it can lead to division by zero. This confirms the correct answer is (a).

Which of the following statements is true regarding Naïve Gaussian elimination?Single choice

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