Exercise 3: Why "Always Pivot and Every System Is Reliable" Is Wrong — Possible Solution
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WHAT PARTIAL PIVOTING ACTUALLY FIXES
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This chapter verified partial pivoting completely resolved the
10^-16-pivot example, dropping x's relative error from about 122%
down to about 10^-16. But that system's underlying problem was
well-conditioned - its true solution was not especially sensitive to
small changes in the input coefficients. The only thing wrong with
it was the ORDER in which Gaussian elimination happened to process
the rows, which produced an unnecessarily huge multiplier. Pivoting
fixes exactly this kind of problem, because it's a property of the
algorithm's own behavior (Chapter 5's territory), and choosing a
different row order is a real, available fix.

WHY THE SECOND WORKED EXAMPLE SHOWS THE LIMIT OF THAT FIX
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This chapter's near-singular matrix example (reused from Chapter 6)
had a condition number of about 40,004 - meaning the TRUE, exact
solution to that system is itself extremely sensitive to small
changes in the input coefficients, independent of how the arithmetic
is organized. Pivoting only changes which row gets used as the pivot
at each step; it does not change the matrix's own determinant,
inverse, or condition number in any way. Applying partial pivoting to
that near-singular system would correctly carry out the elimination
without introducing any unnecessary extra rounding error of its
own - but it still couldn't produce a trustworthy answer, because the
true answer itself moves enormously for a tiny change in the inputs,
and no amount of careful bookkeeping during elimination changes that
underlying fact.

THE CORRECTED CONCLUSION
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Partial pivoting reliably fixes algorithm-level instability - it
should indeed be used essentially always, since it costs little and
never makes things worse. But it does not make every linear system
"solved reliably" in the sense of guaranteeing a trustworthy answer,
because it has no effect on a system's own inherent conditioning.
A complete, honest practice is: always pivot (to eliminate the
algorithm-level risk this chapter demonstrated), AND separately check
the system's condition number (to detect the problem-level risk
Chapter 6 demonstrated) before trusting the result, since these are
two independent risks that require two independent countermeasures.

WHY THIS WORKS AS AN ANSWER
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The explanation credits pivoting with what it genuinely fixes (the
first worked example) rather than dismissing it, then uses this
chapter's own second worked example to show a case pivoting cannot
help with, and arrives at a corrected, more complete recommendation
that treats stability and conditioning as two separate things to
check, matching the chapter's own explicit side-by-side comparison.