Learn Prove factorization conditions, verify row-permutation receipts and reuse exact factors without confusing singularity, free variables or arithmetic cost with numerical stability.
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A factorization records equivalent equations; a permutation receipt PA=LU requires the correspondingly permuted request Pb.
The operation replaces entire equations, including their constants. Applying it only to coefficients generally changes the solution set.
Unpivoted factorization needs stronger intermediate conditions than invertibility of the whole matrix.
No. The matrix with rows (0,1),(1,0) has determinant −1 and is invertible, but its upper-left entry is zero. A row permutation makes a valid first pivot available.
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Exceptional pivots, late swaps and reusable receipts
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Prove the exact no-pivot criterion, recover normalized LDU factors, and diagnose singular and rectangular boundaries.
6 practice tasks.
Track permutations and stored multipliers through late row exchanges, then reuse a verified factorization for new right-hand sides.
6 practice tasks.