Watch a 4×4 matrix reduce to upper-triangular form one row operation at a time — the algorithm behind every linear solve, factorization, and determinant.
Gaussian Elimination StudioLive
LU decomposition with partial pivoting
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Step through real Gaussian elimination. Partial pivoting swaps rows to keep the pivot large (cyan) and stable; the multipliers build L while the reduced rows build U.
Reading this result: Every pivot was already the largest in its column, so no swaps were needed (P = I). The multipliers fill in L, the reduced rows form U, and det(A) = product of U's diagonal = 661.
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How it works
Gaussian elimination with partial pivoting factors a matrix as PA = LU: the permutation P records the row swaps that keep the pivot as large as possible (crucial for numerical stability), the multipliers fill the unit-lower-triangular L, and the reduced rows become the upper-triangular U. Step through each swap and row operation, watch the active pivot highlighted in cyan, and see the determinant emerge as the signed product of U's diagonal. A zero pivot with nothing to swap in signals a singular matrix — no unique factorization, no inverse, det(A) = 0.
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The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.