LU Decomposition Calculator

About this calculator

LU decomposition factors a matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, so that A = LU. This calculator uses the Doolittle method — which fixes L's diagonal entries to 1 and derives the rest through Gaussian elimination — to factor any 2×2 matrix and also reports its determinant, which falls out directly as the product of U's diagonal entries.

Once a matrix is split into triangular factors, solving a linear system Ax = b becomes two simple substitution steps instead of one harder elimination, which is why LU decomposition is a core technique in numerical linear algebra, engineering simulations, and computer graphics — anywhere the same system needs to be solved repeatedly for different right-hand sides, since the expensive factorization only needs to be done once.