Break matrices into lower and upper triangular factors. Follow each pivot step with readable calculations. Practice matrix methods confidently using guided outputs and exports.
Enter a square matrix, choose formatting options, and optionally include vector b to solve a full linear system.
This sample matrix matches the default example and demonstrates how the page can also solve a system when vector b is supplied.
| Row | a₁ | a₂ | a₃ | b |
|---|---|---|---|---|
| 1 | 2 | 3 | 1 | 9 |
| 2 | 4 | 7 | 7 | 23 |
| 3 | -2 | 4 | 5 | 12 |
Without pivoting, the factorization target is A = LU. With pivoting enabled, the target becomes PA = LU.
For Doolittle-style elimination, each multiplier is computed as mᵢₖ = Uᵢₖ / Uₖₖ. The calculator stores that multiplier in Lᵢₖ.
Each lower row is updated with Rᵢ = Rᵢ - mᵢₖRₖ. After all eliminations, the remaining matrix is upper triangular and becomes U.
When vector b is included, the page solves Ly = Pb first, then solves Ux = y to find the final unknown vector x.
It factors a square matrix into a lower triangular matrix and an upper triangular matrix. With pivoting, a permutation matrix is also used, so the relationship becomes PA = LU.
Pivoting swaps rows to avoid zero or tiny pivots. That improves numerical stability and lets the calculator handle more matrices that fail under plain LU decomposition.
Not every matrix has a clean A = LU form without row swaps. Many still decompose successfully when permutation rows are allowed, giving PA = LU instead.
L stores the elimination multipliers below the diagonal. U stores the updated upper triangular coefficients after each elimination step.
Yes. Enter the right-hand-side vector b and the page performs forward substitution and backward substitution after factorization to estimate x in Ax = b.
The determinant comes from the product of U's diagonal entries, adjusted for row-swap parity when pivoting is active.
It compares the factored form with the permuted original matrix. Small error means the computed factors reproduce the input accurately.
Square numeric matrices from 2×2 through 6×6 work best here. Extremely ill-conditioned inputs can still suffer rounding effects.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.