438 ACTIVITY 3: Matrix Row Operations
WHY:
This activity introduces the concept of an elementary row matrix and
illustrates how
these matrices are useful in computer implementation of the Gauss
elimination
process. Row reduction techniques will be used throughout the first
half of the course
when we are studying linear programming and its prerequisites.
Using the computer
to perform repetitive matrix operations removes many of the sources
of error common
to pencil and paper work.
LEARNING OBJECTIVES:
- Discover the role of elementary matrices in row reduction.
- Discover the relationship between elementary matrices and the
inverse of a matrix.
- Provide an opportunity to experience cooperative group learning
using MAPLE.
PERFORMANCE CRITERIA:
- Quality of the answers to the Critical Thinking Questions.
- Completeness
- Clarity
- Depth
-
Understanding how Gauss elimination can be done using matrix
multiplication of elementary
matrices.
- Depth of insight
- Success in completing Maple activity
INFORMATION:
- Matrices:
- A matrix is a set of numbers organized in rows and
columns.
- Matrices as Named Variables:
- You can assign a name to a matrix
and treat the matrix as a variable.
- Identity Matrix:
- This is a special square matrix which is zero
except for ones down the diagonal.
1 0 0 1 0 1 0 0 0
0 1 0 0 1 0 1 0 0
0 0 1 0 0 1 0
0 0 0 1
- Coefficient Matrix for a System of Equations:
- You can write in
matrix form the coefficients of the variables in a system of equations.
- Augmented Matrix:
- You can attach the constants on the right
sides of the equalities of a system of
equations as the last column of the coefficient matrix.
- Elementary Row Operation:
- You can do the same things to the
augmented matrix that you can do to the system of equations:
- Interchange rows
- Multiply a row by a constant, changing that row
- Use a pivot row to change a target row by multiplying the
pivot row by a constant and adding the resulting products
to the target row. The pivot row is not changed.
- Elementary Matrix:
- This is a matrix formed by applying an
elementary row operation to an Identity matrix.
0 1 0 2 0 0 1 0 0
(1) 1 0 0 (2) 0 1 0 (3) 0 1 3 target row
0 0 1 0 0 1 0 0 1 pivot row
Exchange rows 1 & 2 Multiply row 1 by 2 Mult row 3 by 3
and add it to row 2
- Equivalent Matrices:
- Matrices are equivalent if we can change one
into the other by performing row
operations. Equivalent augmented matrices come from systems of equations with the same set of
solutions.
- Gauss-Jordan Elimination:
- This process applies elementary row
operations to an augmented matrix
in an organized way to produce an equivalent matrix whose solution is obvious.
- Inverses:
- The nxn matrix C is the inverse of the matrix D if
CD is the Identity matrix.
- Maple Information:
- Maple commands follow the command prompt >.
- Each command line ends with a semicolon ;(if the results are to be
evaluated and displayed) or a colon :(if the results are not to be
evaluated or not displayed).
- A variable or function definition is indicated by the symbol := (colon
equal).
- A command may be highlighted, edited, and reexecuted (by pressing
Enter). The new results replace the old ones.
- A command may be copied and placed after a new prompt. In this case,
the old results remain on the worksheet at their original location.
- To execute a command, put the cursor anywhere on its line and press
Enter.
RESOURCES:
- Chapter 0, Strategic Mathematics
- MAPLE help system
- Activity2 workspace
- 40 minutes
PLAN:
- Choose roles if you have not already done so.
- Assess the team's understanding of the terms listed in the
information section.
- Execute the model in the Activity2 MAPLE workspace by launching
MAPLE and then
opening Activity2 as Maple text. Have the recorder write down what happened at each step
and any discoveries the team made as the model was executed.
- Use the model to answer the critical thinking questions.
MODEL:
Get into the MAPLE text Activity2 by launching Maple from the Start
menu
and execute
each line. After executing the last A_INV:=multiply(EL,A_INV) command, repeat
the execution of the three commands listed below, changing the
definition of the elementary
matrix EL until AUG is gradually transformed into the Identity matrix augmented by
the following solution vector:
[4 -3 -2]T
EL:= addrow(E,pivotrow,targetrow,-value);
AUG:= multiply(EL,AUG);
A_INV:= multiply(EL,A_INV);
Note that you have to use mulrow instead of addrow to make the pivot element 1.
The problem to be solved in this model is:
x + 2y - 3z = 4
2x + 6y - 8z = 6
3x + 4y - 4z = 8
CRITICAL THINKING QUESTIONS:
These will be provided in class.
SKILL EXERCISES:
- On a piece of paper set up the augmented matrix and perform
Gauss-Jordan elimination to obtain a solution
of the following system: Write down the elementary matrix for each step.
4x + 4y + 2z = 8
x + 2y = 0
3x - y - z = 7
Repeat this process using MAPLE.
- Find and test the inverse of the coefficient matrix of the
system in problem 1.
- If we have the following system of equations and form the matrix
equation as follows:
3x + 5y = 4 |3 5| |x| = |4| D = | 7 -5|
4x + 7y = 5 |4 7| |y| |5| |-4 3|
What should happen if we multiply both sides of the matrix
equation on the left by the matrix D? Try it. Can
you find the solution to this system immediately from what you have just done?
- If you wanted to solve the system
3x + 5y = 7
4x + 7y = 23
Could you do it with one multiply operation? Try it by changing the
commands you used for question 3.
- Check to see if the following matrices are inverses.
1 1 0 1 1 95 -1 -29
2 0 -5 8 0 -92 1 28
0 1 3 2 0 31 0 -10
0 0 3 31 0 -3 0 1
Math 438 Activity 4 -- Revised 7/15/99