Math 410 Second Hour Exam June 18, 1996

Your Name:

There are five questions, each worth 20 points, for a total of 100 points. For full credit you must show all your work.

1. Let A = ( \begin{matrix}−1 & 2 & 0 \\ 1 & −3 & 1 \\ 2 & −2 & 3 \end{matrix} ).

a) Find an LU decomposition of A. (There is a blank page attached for scratch work.)

b) Use your LU decomposition to find all solutions to Ax = b where b = ( \begin{matrix}−1 \\ −2 \\ 3 \end{matrix} ).

2. Let A = ( \begin{matrix}1 & 2 & −1 \\ 2 & 0 & 1 \\ −1 & 1 & t \end{matrix} ) where t is a variable. a) Compute the determinant of A.

b) For which values of t is A invertible?

3. a) Suppose A is a 3 × 3 matrix. You perform the following row operations on A: (i) Multiply row 1 by 1/3 (ii) Subtract 7 times row 1 from row 3 (iii) Switch rows 2 and 3 (iv) Multiply row 2 by 4.

The final result is:

( \begin{matrix}1 & 1 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{matrix} )

What is the determinant of A?

b) Suppose B has the LU decomposition

B = ( \begin{matrix}2 & 0 \\ 5 & −1 \end{matrix} ) ( \begin{matrix}1 & 17 \\ 0 & 1 \end{matrix} ) .

What is the determinant of B?

4. Let A = ( \begin{matrix}1 & 2 & 4 & 2 \\ 1 & 1 & 3 & 2 \\ −1 & 0 & −2 & −2 \end{matrix} ).

a) What is the rank of A?

b) Is the equation Ax = b consistent for all b? Why or why not?

c) Does there exist a b for which Ax = b has a unique solution? Give an example or a reason why no example exists.

5. Use row reduction to compute the inverse of a = ( \begin{matrix}2 & 0 & −1 \\ 0 & 1 & 2 \\ 3 & 1 & 1 \end{matrix} ). YOU MUST SHOW ALL WORK.