Numerical Methods, MAT 350
Autumn 2001

Homework: Some Linear Algebra

Due Friday, November 9, 2001 in class

In the following, you will solve a system of linear equations A x = b using Gauß-Jordan elimination and by using LU factorization. You will also calculate the determinant of A.

This homework set is to be done by hand; you may use a calculator for doing arithmetic. Show your work!

  1. Construct a 4 by 4 matrix A with numerical entries. A should not contain a given number more than once. Construct a length 4 vector b with numerical entries. Each student is to construct a different matrix.
  2. Calculate the solution of A x = b using Gauß-Jordan elimination and back-substitution. It is possible to choose A such that the system has many solutions or no solutions; if this happens to you, you will have to start over with a new matrix.
  3. How many flops does one need to:
    1. get A into upper triangular form?
    2. perform the back-substitution?
    Briefly explain your answers (don't just write your answer down).
  4. Calculate the LU factorization of A.
  5. How many flops does one need to compute the LU factorization?
  6. Find a book on linear algebra and look up the determinant of a matrix.
    1. What is the determinant of the product of two matrices?
    2. What is the determinant of a triangular matrix? You can probably guess the answer by looking at 2 by 2 and 3 by 3 triangular matrices.
    3. Use the LU factorization to calculate the determinant of A. Hint: this should be very easy!
  7. Solve for x using the LU factorization.
  8. Compare the amount of calculational work needed to find x using Gauß-Jordan elimination versus LU factorization.