LU Decomposition Calculator

Determinant and the L times U reconstruction check
Check	Result
det(A) = product of U diagonal	—
L × U reconstructs A	—

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Walk-through

How to Use This Calculator

5 steps

1

Pick a matrix size

Choose 2×2, 3×3, or 4×4 from the size selector on the L & U Matrices tab. The entry grid rebuilds instantly, and any values that still fit the new size are kept.

2

Enter matrix A's entries

Type a number into each cell, row by row. Negative numbers and decimals are both fine — the decomposition recalculates on every keystroke, no Calculate button required.

3

Read L and U

The result card shows the lower-triangular factor L and the upper-triangular factor U as soon as every cell is filled in. If a row swap was needed, the permutation matrix P also appears.

4

Open the Steps tab

Switch to Steps to see the full Gaussian-elimination walk — every pivot search, any row swap, and every elimination that zeroes out an entry below the pivot, each with the matrix state at that point.

5

Check the determinant and verify

Open Determinant & Verify to see det(A) computed as the (sign-corrected) product of U's diagonal, plus a side-by-side check that L × U reconstructs the (possibly row-reordered) original matrix. Use the preset chips to see a no-pivot case, a pivoting case, and a singular case worked out instantly.

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Reference

Formula & Methodology

3 formulas

Doolittle LU factorization

A = L · U (or, with pivoting, P · A = L · U)

Doolittle's method factors a square matrix A into a lower-triangular matrix L with 1s on its diagonal and an upper-triangular matrix U, by running ordinary Gaussian elimination and recording the elimination multipliers into L instead of discarding them. When a pivot (the diagonal entry currently being eliminated below) is too small or zero, the calculator swaps in the largest available entry from the rows below it — a permutation matrix P then tracks which original rows ended up where, so the exact identity that holds is P · A = L · U.

Elimination multiplier (entries of L)

L[i][j] = A[i][j] ⁄ A[j][j] (evaluated at the moment column j is eliminated)

Each sub-diagonal entry of L is the multiplier used to zero out A[i][j] during elimination — the amount of pivot row j subtracted from row i. U is simply the matrix left behind after every elimination step: its diagonal and above-diagonal entries are whatever survives once all the columns to its left have been eliminated.

Determinant from U

det(A) = (−1)^(number of row swaps) · U₁₁ · U₂₂ · ⋯ · Uₙₙ

Because L is unit lower-triangular (det(L) = 1) and the permutation P built from partial pivoting has det(P) = (−1) raised to the number of row swaps performed, det(A) works out to the product of U's diagonal entries, sign-flipped once for every swap. With no pivoting, that sign correction is simply +1.

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Glossary

Key Terms Explained

7 terms

LU decomposition Factoring a square matrix A into a lower-triangular matrix L and an upper-triangular matrix U such that A = LU (or P·A = LU when row swaps were needed). It packages the work of Gaussian elimination so it can be reused efficiently.

Lower triangular A matrix where every entry above the main diagonal is zero. In Doolittle's method, L is additionally unit lower-triangular — its diagonal entries are all 1.

Upper triangular A matrix where every entry below the main diagonal is zero. U is what's left of A after Gaussian elimination has zeroed out everything below each pivot.

Doolittle method The specific LU convention used here: L's diagonal is forced to be all 1s, so every elimination multiplier has a single, unambiguous home in L and U carries whatever scale factor remains. (The alternative, Crout's method, instead forces U's diagonal to be 1s.)

Gaussian elimination The row-reduction process of using one row (the pivot row) to zero out the entries below it in the same column, repeated column by column until the matrix is upper triangular. LU decomposition is Gaussian elimination with the multipliers kept instead of thrown away.

Partial pivoting Before eliminating a column, swapping in whichever remaining row has the largest-magnitude entry in that column, so it becomes the pivot. This avoids dividing by a zero (or a needlessly small) pivot and keeps the arithmetic numerically stable.

Matrix factorization Rewriting a matrix as a product of two or more simpler matrices. LU decomposition is one of the most widely used factorizations because triangular systems (like L and U) are far cheaper to solve than a general matrix.

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Scenarios

Real-World Examples

3 worked examples

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3×3 without pivoting

A = [[4, 1, 1], [2, 5, 1], [1, 1, 3]]

Size 3×3 Matrix [4, 1, 1, 2, 5, 1, 1, 1, 3]

Column 1's largest entry is already on the diagonal (4 beats 2 and 1), so no swap is needed. Eliminating gives L = [[1,0,0],[0.5,1,0],[0.25,0.1667,1]] and U = [[4,1,1],[0,4.5,0.5],[0,0,2.6667]]. det(A) = 4 × 4.5 × 2.6667 = 48, matching a direct expansion.

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Matrix needing partial pivoting

A = [[0, 1, 1], [2, 1, 1], [1, 2, 3]]

Size 3×3 Matrix [0, 1, 1, 2, 1, 1, 1, 2, 3]

A zero sits in the (1,1) spot, so column 1 immediately swaps in row 2 (entry 2). After eliminating, column 2 needs a second swap. Two swaps means the sign correction cancels out: det(A) = +(2 × 1.5 × −0.6667) = −2, which matches expanding the original A directly.

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Determinant read off U's diagonal

A = [[2, 1, 0, 3], [0, 3, 4, 1], [0, 0, 1, 2], [0, 0, 0, 5]]

Size 4×4 Matrix [2, 1, 0, 3, 0, 3, 4, 1, 0, 0, 1, 2, 0, 0, 0, 5]

A is already upper triangular, so U = A exactly and L is the identity — no elimination or pivoting to do. det(A) is simply the product of the diagonal: 2 × 3 × 1 × 5 = 30.

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Deep Dive

Understanding the LU Decomposition Calculator

3 chapters · 4 min read

LU decomposition rewrites a square matrix A as the product of a lower-triangular matrix L and an upper-triangular matrix U, so that A = LU. It's the same arithmetic as ordinary Gaussian elimination, just packaged so the elimination work can be reused — which is exactly why it's the default method most numerical libraries use to solve linear systems, invert matrices, and compute determinants.

Why bother factoring A at all?

If you only need to solve Ax = b once, plain Gaussian elimination on the augmented matrix works fine. LU decomposition pays off when you need to solve Ax = b for the same A against many different b vectors — a common situation in simulations, circuit analysis, and iterative solvers. Once A = LU is computed, each new b only costs a cheap forward-substitution (solve Ly = b) followed by a cheap back-substitution (solve Ux = y), instead of re-running the full O(n³) elimination from scratch every time.

How partial pivoting keeps it correct

Gaussian elimination breaks down the moment a pivot (the diagonal entry you're about to divide by) is zero — you can't divide by zero, and dividing by a very small number amplifies rounding error. Partial pivoting fixes this by scanning the remaining rows in the current column and swapping in whichever one has the largest absolute value before eliminating. Every swap is tracked in a permutation matrix P, so the identity that holds afterward is P·A = L·U rather than A = L·U. This calculator's Steps tab shows exactly when and why each swap happens.

The determinant is a free by-product

Because L is triangular with 1s on its diagonal, det(L) = 1 always. Because U is triangular, its determinant is just the product of its diagonal. And because each row swap flips the sign of a determinant, det(A) works out to (−1)^(number of swaps) × (product of U's diagonal) — no separate cofactor expansion required. This is also why a zero anywhere on U's diagonal signals a singular matrix: the product becomes zero, so det(A) = 0 no matter what pivoting did along the way.

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Questions

Frequently Asked Questions

6 questions

What's the formula behind LU decomposition?+

A = LU, produced by running Gaussian elimination and recording each elimination multiplier into L instead of discarding it, while U keeps whatever remains after elimination. When partial pivoting is needed, the exact relationship is P·A = LU, where P records any row swaps.

What does "Doolittle" mean?+

Doolittle's method is the specific LU convention this calculator uses: L's diagonal entries are all forced to be 1, so each elimination multiplier has one unambiguous place to live. (Crout's method is the mirror-image convention, forcing U's diagonal to be 1s instead.)

When does LU decomposition fail?+

It fails outright only when a column has no usable pivot — every entry from the current row down is zero — and no row swap can fix that. Partial pivoting (swapping in the largest available entry, giving P·A = LU) rescues most cases where the raw diagonal entry alone would have been zero.

How do you get the determinant from an LU decomposition?+

Multiply U's diagonal entries together, then flip the sign once for every row swap performed during partial pivoting. With no swaps, det(A) is simply the product of U's diagonal.

What is LU decomposition actually used for?+

Mainly for solving Ax = b efficiently, especially when the same A needs to be solved against many different b vectors — each new solve only costs a forward- and back-substitution instead of a full elimination. It's also the standard method numerical libraries use internally to compute determinants and matrix inverses.

How can I check that my L and U are correct?+

Multiply L by U and compare the result to the original matrix A (or to P·A if any row swaps occurred) — they should match. This calculator does exactly that check automatically on the Determinant & Verify tab.

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Matrix A

A = L × U

Gaussian elimination steps

Elimination walk

det(A) and verification

det(A)

How to Use This Calculator

Pick a matrix size

Enter matrix A's entries

Read L and U

Open the Steps tab

Check the determinant and verify

Formula & Methodology

Key Terms Explained

Real-World Examples

3×3 without pivoting

Matrix needing partial pivoting

Determinant read off U's diagonal

Understanding the LU Decomposition Calculator

Why bother factoring A at all?

How partial pivoting keeps it correct

The determinant is a free by-product

Frequently Asked Questions

Get a branded PDF of your results

LU Decomposition Calculator

Matrix A

A = L × U

Gaussian elimination steps

Elimination walk

det(A) and verification

det(A)

How to Use This Calculator

Pick a matrix size

Enter matrix A's entries

Read L and U

Open the Steps tab

Check the determinant and verify

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the LU Decomposition Calculator

Why bother factoring A at all?

How partial pivoting keeps it correct

The determinant is a free by-product

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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