##
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!

- 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.
- 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.
- How many flops does one need to:
- get
**A** into upper triangular form?
- perform the back-substitution?

Briefly explain your answers (don't just write your answer down).
- Calculate the LU factorization of
**A**.
- How many flops does one need to compute
the LU factorization?
- Find a book on linear algebra and look up the
determinant of a matrix.
- What is the determinant of the product of two matrices?
- 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.
- Use the LU factorization to calculate the determinant of
**A**.
Hint: this should be very easy!

- Solve for
**x** using the LU factorization.
- Compare the amount of calculational work needed to find
**x** using Gauß-Jordan elimination versus LU
factorization.