Question

Determine the LU-Decomposition of the following matrix:

A = [-4 1 -5 -15; 1 2 -3/2 1; -3 3 -12 3/2; 2 1 2 1]

Answer 1

The LU-Decomposition of the matrix A is L = [1 0 0 0; -⅓ 1 0 0; -⅓ 0 1 0; -⅓ 0 -⅓ 1] and U = [-4 1 -5 -15; 0 2 -⅓ 5; 0 0 -⅓ 0; 0 0 0 -2].

Step-by-step explanation:

The LU-Decomposition of the matrix A can be determined using Gaussian elimination. Firstly, find the L matrix by performing row operations to eliminate the coefficients below the main diagonal, resulting in:

A1 = [-4 1 -5 -15]

A2 = [0 2 -⅓ 5]

A3 = [0 0 -⅓ 0]

A4 = [0 0 0 -2]

The U matrix can be obtained by performing row operations on the original matrix A, resulting in:

U = [-4 1 -5 -15]

U = [0 2 -⅓ 5]

U = [0 0 -⅓ 0]

U = [0 0 0 -2]

Therefore, the LU-Decomposition of matrix A is:

L = [1 0 0 0;
-⅓ 1 0 0;
-⅓ 0 1 0;
-⅓ 0 -⅓ 1]

U = [-4 1 -5 -15;
0 2 -⅓ 5;
0 0 -⅓ 0;
0 0 0 -2]