Question

(3 points) Find the LDU factorization of 「-3-15 15 A=1 12 64-68 -9 -37 26 That is, write A = LDU where L is a lower triangular matrix with ones on the diagonal, D is a diagonal matrix, and U is an upper triangular matrix with ones on the diagonal.

Answer 1

A = [-3 -15 15 12 64 -68 -9 -37 26] Let L = [1 0 0 l_21 1 0 l_31 l_32 1] , U = [1 u_12 u_13 0 1 u_23 0 0 1] , D = [d_1 0 0 0 d_2 0 0 0 d_3] Now LDU = [1 0 0 l_21 1 0 l_31 l_32 1] [d_1 0 0 0 d_2 0 0 0 d_3] [1 u_12 u_13 0 1 u_23 0 0 1] = [d_1 0 0 l_21 d_1 d_2 0 l-31 d_1 l_32 d_2 d_3] [1 u_12 u_13 0 1 u_23 0 0 1] = [d_1 u_12 d_1 d_1 u_13 l_21 d_1 l_21 u_12 d_1 l_21 d_1 u_13 + d_2 u_23 l_31 d_1 l_31 d_1 + l32 d_2 l_31 d_1 + l_32 d_2 u_23 + d_3] As, we have A = LDU On comparing the elements of both matrices. \r\n d-1 = -3, u_12 d_1 = -15 implies u_12 = 5,