Exceptional pivots, late swaps and reusable receipts
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Work over the real numbers in these changed cases, using exact rational expressions. Verify factor identities and solutions in the original equations.
State normalization, pivot and cost assumptions. Public worked solutions support practice; saved proofs are not independently verified examinations.
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Let A(t) have rows (t,1,0),(1,1,1),(0,1,2), with real t. Determine exactly when A(t)=LU with L unit lower triangular and U upper triangular without row permutations, including singular t. Give the factors where they exist and prove nonexistence at exceptional parameters. Obtain LDU with unit lower and unit upper factors, distinguishing invertible D from a diagonal factor allowed to have a zero. Compute all leading principal minors and explain why an invertible A can nevertheless fail the unpivoted factorization.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For A with rows (2,1,1),(4,2,3),(6,5,4), eliminate the first column, then resolve the next zero pivot by swapping the remaining rows. Construct P,L,U with PA=LU and unit lower L. Explain why the already stored multipliers must move in that swap. Use the same factors to solve Ax=b for b=(3,8,9) and b=(1,1,6), showing Pb, the intermediate solution of Ly=Pb and the final solution of Ux=y. Verify both in the original equations. Diagnose the receipt obtained by leaving the first-column multipliers in their pre-swap order.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let A have three rows (1,2,0,1),(2,4,1,3),(3,6,1,4). Verify a factorization A=LU with L unit lower triangular of size 3 and U row echelon of size 3×4. For arbitrary real b, characterize consistency and every solution. Give one consistent changed request and one inconsistent request, and explain why reusing factors does not imply a unique triangular solve. Separately, for a dense nonsingular n×n system with a fixed valid pivoted factorization and s right-hand sides, justify the conventional arithmetic-work comparison Θ(n³+s n²) versus Θ(s n³), stating the model and excluded costs. Do not apply the nonsingular solve claim to the rectangular example.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let A(t) have rows (t,1,0),(1,1,1),(0,1,2), with real t. Determine exactly when A(t)=LU with L unit lower triangular and U upper triangular without row permutations, including singular t. Give the factors where they exist and prove nonexistence at exceptional parameters. Obtain LDU with unit lower and unit upper factors, distinguishing invertible D from a diagonal factor allowed to have a zero. Compute all leading principal minors and explain why an invertible A can nevertheless fail the unpivoted factorization.
For A with rows (2,1,1),(4,2,3),(6,5,4), eliminate the first column, then resolve the next zero pivot by swapping the remaining rows. Construct P,L,U with PA=LU and unit lower L. Explain why the already stored multipliers must move in that swap. Use the same factors to solve Ax=b for b=(3,8,9) and b=(1,1,6), showing Pb, the intermediate solution of Ly=Pb and the final solution of Ux=y. Verify both in the original equations. Diagnose the receipt obtained by leaving the first-column multipliers in their pre-swap order.
Let A have three rows (1,2,0,1),(2,4,1,3),(3,6,1,4). Verify a factorization A=LU with L unit lower triangular of size 3 and U row echelon of size 3×4. For arbitrary real b, characterize consistency and every solution. Give one consistent changed request and one inconsistent request, and explain why reusing factors does not imply a unique triangular solve. Separately, for a dense nonsingular n×n system with a fixed valid pivoted factorization and s right-hand sides, justify the conventional arithmetic-work comparison Θ(n³+s n²) versus Θ(s n³), stating the model and excluded costs. Do not apply the nonsingular solve claim to the rectangular example.
