3×3 LU Decomposition Calculator
LU decomposition of a 3×3 matrix with steps. Get the lower triangular L and upper triangular U, the 3 elimination multipliers, and a check that LU returns your original matrix.
Factor a 3×3 matrix into L and U. The grid is preset to 3×3, and the 3 multipliers that build L are named as each one is used.
The shape of L and U at 3×3
L is 3×3 with 1s down the diagonal, zeros above it, and 3 multipliers below. U is 3×3 with zeros below the diagonal and the eliminated values above. Between them they hold exactly the same information as the original matrix — 9 numbers in, 9 numbers out.
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Frequently asked questions
How many multipliers does a 3×3 LU decomposition have? ▾
A 3×3 factorisation has 3 multipliers — one for each entry below the diagonal of L. The rest of L is 1s on the diagonal and 0s above it.
How do you check a 3×3 LU decomposition? ▾
Multiply L by U. If the product is your original matrix, the factorisation is right. The calculator performs this check and shows the product for you.
What if a pivot is zero in a 3×3 matrix? ▾
Elimination cannot divide by zero, so the rows get swapped and the swap is recorded in a permutation matrix P. The result is then PA = LU, which the calculator labels clearly.
About this Calculator
LU decomposition of a 3×3 matrix with steps. Get the lower triangular L and upper triangular U, the 3 elimination multipliers, and a check that LU returns your original matrix.