LU Factorization Calculator
LU factorization calculator using the Doolittle method, with every multiplier shown. Factor a square matrix as A = LU (or PA = LU with pivoting) and use it to solve systems and find determinants.
A = LU by the Doolittle method, with every multiplier named as it is computed. Enter any square matrix and the calculator returns L, U, the determinant that falls out of U's diagonal, and a verification that LU rebuilds your matrix.
How to find the LU factorization of a matrix
- 1
Start with L = I and U = A
The identity for L and a copy of your matrix for U.
- 2
Compute each multiplier
For the entry in row i, column j below the diagonal, the multiplier is that entry divided by the pivot in row j.
- 3
Eliminate and record
Subtract the multiplier times the pivot row from row i in U, and store the multiplier at L[i][j].
- 4
Verify
Multiply L and U — the result must equal the original matrix (or PA if rows were swapped).
Doolittle versus Crout
| Method | Diagonal of 1s | Typical use |
|---|---|---|
| Doolittle | In L | The default in most courses, and what this calculator produces |
| Crout | In U | Common in older numerical libraries and some engineering texts |
| Cholesky | Neither — U = Lᵀ | Symmetric positive-definite matrices only, at roughly half the cost |
LU factorization by size
Frequently asked questions
Is LU factorization the same as LU decomposition? ▾
Yes — the two names describe the same thing, writing a square matrix as a lower triangular matrix times an upper triangular one. American textbooks tend to say factorization; numerical analysis literature tends to say decomposition.
What is the Doolittle method? ▾
Doolittle is the version that fixes 1s along the diagonal of L. Each multiplier from forward elimination drops into the position of the entry it cleared, and U is whatever elimination leaves behind. This calculator uses Doolittle.
When does LU factorization fail? ▾
It fails without pivoting whenever a pivot position holds a zero, which happens when a leading principal minor is zero. Allowing row swaps always fixes it, at the cost of the permutation matrix P in PA = LU.
How do you use LU factorization to solve Ax = b? ▾
Substitute A = LU to get LUx = b. Set y = Ux and solve Ly = b by forward substitution, then solve Ux = y by back substitution. Each solve is quick because both matrices are triangular.
How does LU factorization find a determinant? ▾
det(A) = det(L)det(U). Since L is unit triangular its determinant is 1, so det(A) is just the product of U's diagonal, with a sign flip for each row swap.
What is the computational cost? ▾
Factoring an n x n matrix takes about 2n³/3 operations, the same as one Gaussian elimination. The payoff comes afterwards: every additional right-hand side costs only about 2n² operations.
About this Calculator
LU factorization calculator using the Doolittle method, with every multiplier shown. Factor a square matrix as A = LU (or PA = LU with pivoting) and use it to solve systems and find determinants.