# Math 5334: Homework 2 Solutions

Note: The material you hand in should be legible (either typed or neatly
hand-written),

well-organized and easy to mark, including the use of good English. All computer
programs

should be handed in and should be well commented. In general, short, simple
answers are

worth more than long, complicated ones. Unless stated otherwise, all answers
should be

justified.

1. The following exercises from Chapter 2 in the text are to be done by hand.

(a) Question **2.61** on page 95.

(b) Question **2.65** on page 96.

(c) Question **2.4** on page 97.

(d) Question **2.7** on page 97.

(e) Question** 2.12 **on page 97.

(f) Question** 2.17** on page 97.

(g) Question** 2.21** on page 98.

2. The following computer problems are to be done using Matlab or Octave. For
each

question, hand in your program code and a transcript of a terminal session
demonstrating

that your programs work correctly. Be sure to indicate clearly which questions

the programs and the transcripts refer to.

(a) (10 points total) Question **2.2** on page 100 (3 points for part a, 2
for b, 5 for

c). You should use the Matlab function lu to compute the LU factorization of

matrix A, and then use the Matlab backslash operator, \, to solve the resulting

triangular systems . (For more details, see "LU Factorization" in the Mathematics

section of the Matlab help pages.)

(b) Question 2.3 on page 100.

(c) Do the following:

i. Write a simple program that, given a square matrix , A, computes the elementary

elimination matrices M_{1}, M_{2} and M_{3}. You do not need to worry about

the efficiency of this program. Just keep it simple .

ii. Test your program on a random 4*4 matrix, A_{1}, and on a random
8*8

matrix, A_{2}. You can use the Matlab function rand to generate a random

matrix.

iii. For both matrices A_{1} and A_{2},
use matrix multiplication to compute the products

M_{1}M_{2}M_{3} and M_{3}M_{2}M_{1}.
Compare the products to the individual matrices M_{1}, M_{2} and M_{3}.

iv. For both matrices A_{1} and A_{2}, use standard matrix
operations to compute

(M_{3}M_{2}M_{1})^{-1}. Compare the results to the individual
matrices M_{1}, M_{2} and

M_{3}. You should use the Matlab function inv to compute the inverse.

v. Write a faster program to compute (M_{3}M_{2}M_{1})^{-1}.
Verify that your program

gives the same results as in part iv.

vi. Generate random n*n matrices for n = 10, 100, 1000,
2000, 3000, 4000, 5000.

Use the Matlab functions tic and toc to measure how much time your pro-

gram in part v takes to execute on each of these matrices. Also measure the

amount of time used by the more-straightforward method in part iv. (If your

computer cannot handle the larger matrices, don't worry about it.)

**No more questions will be added **

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