Answer Happy

In this problem we investigate and use the LU decomposition of a matrix A (the process of writing A = LU) where L is lower triangular matrix and U is upper triangular. Not every matrix has an LU decomposition, so we are assuming the matrix A has one. Motivation: The LU factorisation often helps to quickly solve equations of the form Ax = B where x and B are column vectors. The system Ax = B can be solved as in two stages as follows: (i) First solve Ly = B for Y for a column vector y. (ii) Then solve Ux=Y for x. (iii) Then x is a solution to Ax = B because Ax = (LU)x=L(Ux) = Ly = B. LU Algorithm: To find the LU decomposition, follow the algorithm below. Let A be an m x n matrix and suppose A can be reduced to a row-echelon matrix U without row exchanges. Then A = LU for some lower triangular matrix L, and matrices U and L are constructed as follows. Step 1: If all the entries of A are zero, namely A = 0, then take L = Im, where Im is the identity matrix of size m, and take U = A. Step 2: If not all the entries of A are zero, namely A #0, then use Gaussian elimination on A to obtain a row-echelon matrix U using no row exchanges. Step 3: Let L be a lower triangular and m m m matrix, namely 0 0 0 0 12.1 12.2 0 0 0 L= : Im-1,1 lm-1.2 Im3 Im-1,m-1 0 lm, 1 Im.2 m3 Im.m. Multiply L and U, and then use A = LU to obtain equations that will give values to lij with i = 1, ..., m and j = 1, ...,i. Here you know the all the entries of A and U and must find the entries (the variables of the equations) of L. 11,1 Im.m-1

Answer the following questions. [2 0 0 2 (a) Let Aj = 0 0 -1 -2 Find an LU decomposition of the matrix by 1 0 1 6 following the LU algorithm above, namely find appropriate matrices Lị and U such that A1 = LU. (b) Verify that A1 = L U, by performing matrix multiplication. (c) Use the decomposition LiU of Aj to solve the system 11 1 C 2 Ajx = B, where x = - B 13 2 14 (d) Find a 2 x 3 matrix A2 that does not have an LU decomposition and justify your answer.