Math 410 First Hour Exam June 11, 1996

Your Name:

There are six questions for a total of 100 points. The numbers in parentheses indicate the points for each question. If you want any partial credit, you must show your work.

1. (20) For each of the systems of equations below, decide whether it has no solutions, a unique solution, or infinitely many solutions. Find all solutions. a)

2x_{1} − 6x_{2} = 0
7x_{1} + 4x_{2} = 25
x_{1}  = 3

b)

2x_{1} + 7x_{2} + x_{3} = 1
−6x_{1} + 4x_{2}  = 2

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

a) Write A as the sum of a symmetric matrix and a skew symmetric matrix.

b) Write A as the sum of an upper-triangular matrix and a lower-triangular matrix.

3. (15) The table below lists the amounts of certain nutrients per serving for two popular cereals. Suppose you want to prepare a mixture of these two cereals which contains exactly 295 calories, 9 grams of protein, 48 grams of carbohydrate, and 8 grams of fat. Decide whether this is possible, and if so what is the desired mix of cereals.

Nutrient  Cheerios  Wheaties
Calories  110  130
Protein (grams)  4  3
Carbohydrates (grams)  20  18
Fat (grams)  2  5

4. (20) Let v_{1} = [1, 2, −1], v_{2} = [0, −2, 2], v_{3} = [2, 1, 1], and v_{4} = [3, 1, 2].

a) Are v_{1}, v_{2}, v_{3}, and v_{4} linearly independent?

b) Do v_{1}, v_{2}, v_{3}, and v_{4} span R^{3}?

c) Do v_{1} and v_{4} span R^{3}?

5. (20) Matrix Town produces equipment and clothing. Producing one dollar worth of equipment requires $.50 worth of equipment and $.25 worth of clothing. Producing one dollar worth of clothing requires $.40 worth of equipment. Also, Matrix town exports $10,000 worth of equipment and it imports $3,000 worth of clothing.

a) Set up a system of equations describing this economy according to the Leontief Input-Output model. (Be careful about rows vs. columns and about signs.)

b) Solve this system of equations to determine the optimal levels of production (in dollars).

6. (15) Let

A = ( \begin{matrix}1 & 0 & 25 & 1 & 1 \\ 0 & 1 & 0 & 5 & −6 \\ 0 & 0 & 10 & −17 & 1 \\ 0 & 0 & 5 & 4 & 1 \\ 0 & 0 & 12 & 13 & 1 \end{matrix} ) .

Decide which entry of A will be the next pivot under each of these methods of row reduction:

a) partial pivoting

b) scaled pivoting

c) complete pivoting.