LU Decomposition Calculator

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Determinant of A
28

det(A) computed as the product of the diagonal entries of U (with sign from row swaps).

Partial pivotingYes - rows were swapped (partial pivoting applied)
Lower triangular L[1, 0, 0]; [0.25, 1, 0]; [0.5, 0.666667, 1]
Upper triangular U[8, 7, 9]; [0, -0.75, -1.25]; [0, 0, -4.666667]
Permutation matrix P[0, 0, 1]; [1, 0, 0]; [0, 1, 0]
SingularityInvertible
Determinant28

PA = LU factorisation complete. Determinant = 28.

  • The determinant is 28, so the matrix is invertible and Ax = b has a unique solution for any b.
  • You can now solve Ax = b efficiently by forward-substituting into Ly = Pb, then back-substituting into Ux = y.
  • Partial pivoting was applied: rows were reordered to put the largest pivot in the lead position, which avoids division by small numbers and reduces floating-point error.
  • The L matrix has 1s on its diagonal and 3 sub-diagonal entries. The U matrix carries the pivots on its diagonal.

Next stepUse the L and U matrices to solve a system: forward-substitute into Ly = Pb, then back-substitute into Ux = y.

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