3.10. This problem considers the question, why not use UL instead of LU? You can assume that A is a 2 ×2 matrix. (a) Find a Doolittle version of the factorization A = UL, where L is a unit lower triangular matrix and U is upper triangular. You can assume pivoting is not necessary. (b) Describe the resulting algorithm for solving Ax = b. (c) Is there any connection between the Doolittle factorization of AT and the one you found in part (a)?

a

The Doolittle version of factorization A =UL is

b

           Algorithm

1)  Obtain the values of the individual elements of U and L

2) Obtain the value of z for the expression 1.e

 3   Obtain the value of x for the expression 1.e

c

  Let say that   is the Doolittle factorization of

where L is a unit lower triangular matrix and U is upper triangular matrix

     Hence

Here   is a unit upper  triangular matrix and lower is triangular  

In the question that we are given  A = UL meaning that   looking at this  U is upper triangular matrix and L is unit lower triangular matrix. So we can say that the Doolittle version given behave like Doolittle factorization only for A and there is no connection with

Step-by-step explanation:

From the question let assume that          

The objective is to find the Doolittle version of the factorized  A = UL and this can be expressed in matrix as

Looking at this in form of matrix multiplication

           =>      

    and          

   Hence            

 b)   From the question we are given

And objective is to describe the algorithm to solve it

Let assume that

               So

   Given that  U is upper triangular  matrix then y can be obtained by back substitution

       Now that we are able to obtain y, given that  L is a unit lower triangular matrix we can obtain x by forward substitution

The algorithm is given in the answer section

28

step-by-step explanation:

because that's the answer