LU Decomposition

About this calculator

The LU Decomposition Calculator factors a square matrix as P·A = L·U using Doolittle elimination with partial pivoting: P is a permutation matrix, L is unit lower-triangular, and U is upper-triangular. All arithmetic runs in 30-digit decimal precision so rational inputs stay exact. It reports the row swaps, the sign (−1)^swaps, the determinant from the diagonal of U, and the verification product L·U. Unlike inversion, singular matrices still factor — a zero pivot simply leaves 0 on U's diagonal and det = 0.

How to use the LU Decomposition calculator

1. Enter the values for your problem into the input fields.
2. Read the result — it updates instantly as you type.
3. Check the formula and the visual explanation to follow how the answer was found.
4. Copy the page URL to share the exact calculation.

Common examples

• [[4,3],[6,3]] → one pivot swap, U = [[6,3],[0,1]], L₂₁ = 0.666…, det = −6
• 3×3 [[2,1,1],[4,−6,0],[−2,7,2]] → U = [[4,−6,0],[0,4,1],[0,0,1]], det = −16
• Singular [[1,2],[2,4]] → U = [[2,4],[0,0]], zero pivot on the diagonal, det = 0
• [[0,1],[1,0]] → pure permutation: one swap, sign −1, det = −1