Ludecomposition Calculator
Free Ludecomposition Calculator for linear algebra. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
A = L * U
Where A is the original square matrix, L is a lower triangular matrix with ones on its diagonal, and U is an upper triangular matrix. The diagonal of U contains the pivot elements, and the product of the pivots gives the determinant.
Worked Examples
Example 1: 3x3 LU Decomposition
Problem:Find the LU decomposition of A = [[2, 1, 1], [4, 3, 3], [8, 7, 9]].
Solution:Step 1: U row 1 = [2, 1, 1]\nStep 2: L column 1 = [4/2, 8/2] = [2, 4]\nStep 3: U[1][1] = 3 - 2*1 = 1, U[1][2] = 3 - 2*1 = 1\nStep 4: L[2][1] = (7 - 4*1)/1 = 3\nStep 5: U[2][2] = 9 - 4*1 - 3*1 = 2\nL = [[1,0,0],[2,1,0],[4,3,1]], U = [[2,1,1],[0,1,1],[0,0,2]]
Result:det(A) = 2*1*2 = 4 | L and U verified: L*U = A
Example 2: 2x2 LU Decomposition
Problem:Find the LU decomposition of A = [[3, 4], [6, 5]].
Solution:U[0][0] = 3, U[0][1] = 4\nL[1][0] = 6/3 = 2\nU[1][1] = 5 - 2*4 = -3\nL = [[1, 0], [2, 1]]\nU = [[3, 4], [0, -3]]\nVerify: L*U = [[3,4],[6,5]] = A
Result:det(A) = 3*(-3) = -9 | Decomposition verified
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy