Module 2: System of Linear Equations and Matrices

Week 2 - Homework Assignment: Matrix Operations

Submissions accepted: 6/27/2012 | 12:00 AM Review: Full, Anonymous: No

Complete the following problems using your textbook:

2.2 Exercise Page 86------2, 14, 22 and 28

2.4 Exercise Page 108------8, 16 and 36

2.5 Exercise Page 116------12 (Page 120 on eBook)

2.2 Exercise Page 86------2, 14, 22 and 28

In Exercises 1 –4, write the augmented matrix corresponding to each system of equations.

2. 3x + 7y – 8z = 5

x + 3z = -2

4x - 3y = 7

Solution:

The augmented matrix will be

3 7  8 5    1 0 3  2 4  3 0 7 

In Exercises 9–1 8, indicate whether the matrix is in row-reduced form

Solution: This matrix is not in row-reduced form since first nonzero entry in row 3 is a 2, not a 1.

In Exercises 19–26, pivot the system about the circled element.

Solution:

Perform the operation R2 – 2R1

The matrix changes to

1 3 4    0  2  2

Answer:

1 3 4    0  2  2

In Exercises 27–30, fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices.

R2 – 2R1 – R2 R1 – 2R2

Solution:

R2 – 2R1 – R2 R1 – 2R2

2.4 Exercise Page 108------8, 16 and 36

In Exercises 7–1 2, refer to the following matrices:

A = B = C = D =

8. Explain why the matrix A + C does not exist. A + C does not exist since dimension of matrix A is not equal to the dimension of matrix C.

In Exercises 1 3–20, perform the indicated operations.

16. 3 + 4

Solution:

 3 3  9  8  4 32       9 6 9   16 8 8  21  3 18  12 24 12 3  (8) 3  (4)  9  32     9 16 6  8 9  8   2112  3  24 18 12   5 1 23    25 14 17 33 21 30

 5 1 23   Answer: 25 14 17 33 21 30

36. INVESTMENT PORTFOLIOS The following table gives the number of shares of certain corporations held by Leslie and Tom in their respective IRA accounts at the beginning of the year:

IBM GE Ford Wal-Mart Leslie 500 350 200 400 Tom 400 450 300 200

Over the year, they added more shares to their accounts, as shown in the following table:

IBM GE Ford Wal-Mart Leslie 50 50 0 100 Tom 0 80 100 50

a. Write a matrix A giving the holdings of Leslie and Tom at the beginning of the year and a matrix B giving the shares they have added to their portfolios.

500 350 200 400 A    400 450 300 200

50 50 0 100 B     0 80 100 50 

b. Find a matrix C giving their total holdings at the end of the year.

C  A  B 500 350 200 400 50 50 0 100       400 450 300 200  0 80 100 50  550 400 200 500    400 530 400 250

550 400 200 500 Answer: C =   400 530 400 250

2.5 Exercise Page 116------12

In Exercises 7–24, compute the indicated products.

Solution:

 1*1 3*3 1*3  3*0 1*0  3* 2     1*1 2*3 1*3  2*0 1*0  2* 2 10 3 6     5  3 4

10 3 6 Answer:    5  3 4

22. What is the difference between scalar multiplication and matrix multiplication? Give examples of each operation.

Solution:

In scalar multiplication, a number (scalar) is multiplied to a matrix and in matrix multiplication, two matrices are multiplied.

Example of scalar multiplication:

10 3 6 20 6 12 2      5  3 4 10  6 8 

Example of matrix multiplication:

10 3 6    5  3 4

 1*10  3*5 1*3  3*(3) 1*6  3* 4     1*10  2*5 1*3  2*(3) 1*6  2* 4 25  6 18     0  9 2 