Question

4. Suppose that A Rnn is nonsingular. We can pose the problem of finding A-1 as the system of linear equations where X e R" is the unknown inverse matrix. We assume that A has LU factorization A LU

Answer 1

Let a square matrix A can be factorize the form LU when L is lower triangular and & U is upper triangular, if all the principle minors of A are singular i.e, it is a standard of linear algebra that such a factorization, when it is exists, is unique. We consider for definiteness the linear system a_11 x_1 + a_12 x_2 + a_13 x_3 + ...... + a_n x_n = b_1 a_21 x_1 + a_22 x_2 + a_23 x_3 + .... + a_2n x_n = b_2 a_31 x_1 + a_32 x_2 + a_33 x_3 + .... + a_nn x_n = b_n which can be written in the form AX = B -(1) Let A = LU -(2) Where L = [1 l_21 l_31 l_n1 0 l_22 l_32 l_m 0 0 l_33 l_n3 0 0 0 1] -(3) and U = [U_11 0 0 0 U_12 U_22 0 0 U_3 U_23 U_33 0 U_1n U_2n U_3n U_nn] -(4) \r\n