Objective:
The computational power of modern software packages and graphing
calculators can make life much easier for students in an introductory
linear algebra class, but this demo illustrates that these powerful
tools can also make life more miserable. This demo shows the power
of modern technology (the good news) but it clearly demonstrates the
need for good, old-fashioned brain power and common sense (the bad news
for students).
Level:
This demo can be presented in connection with any introduction to matrix
inverses via row operations.
Prerequisites: Students should be familiar
with matrix multiplication, elementary row operations that lead to a
reduced row echelon form, augmented matrices, and the definition of
inverse of a matrix.
Platform:
Although the demo description illustrates the use of MATLAB for
matrix computations, the actual demo is platform independent. It
can be used with any computational system that is capable of obtaining
the reduced row echelon form of a matrix and performing matrix
multiplication.
Instructor's
Notes: I use MATLAB in my sophomore level Linear Algebra
course but I defer using it until students have mastered the
row-reduction process by hand for small systems of linear
equations. After that I introduce them to the powerful command
rref, which effortlessly computes the reduced row echelon form of
a matrix. Although students think that solving the systems of
equations is trivial using MATLAB and that there is no need to bother
with extraneous material, I insist that "it is still necessary to
understand the underlying process of solving linear systems."
We enunciate a three-step strategy for
solving most problems in linear algebra using MATLAB.
-
augment
-
rref
-
interpret
When we cover the topic of inverse of a
matrix, our three-step strategy makes the procedure seem trivial.
To find the inverse of a matrix A, the students quickly learn to
augment A with an appropriately sized identity matrix,
rref the augmented matrix, and then examine the right side of the
reduced row echelon form of the augmented matrix to immediately obtain
the matrix inverse.
A typical example follows. To avoid
somewhat daunting decimal display, we change to a rational display
format using the command format rat.
Step
1. Define A and augment with the identity
matrix.
Step
2. rref the augmented matrix.
Step
3. Read the matrix inverse from the right 3 by 3
block. The commands to extract the desired matrix are shown
below.
MATLAB also makes it easy to check to see
that the inverse property is satisfied by the matrix
X:
Note that MATLAB uses floating
point arithmetic; in format rat the symbol * means that no rational
number was found to represent the number. In this context, * means that
the number is very close to zero. With that in mind, we have now
shown that A*X = X*A = I which
indicates that X is the inverse of A.
At this point, armed with their
new tool for easily finding matrix inverses, the students are ready to
end class. However, I have them do one more example. This time,
they replace the (3,3) entry in A by 9 and repeat the same steps
in MATLAB.
Now, you and I know the meaning of the
result from rref(aug), but none of my students seem to.
They have put their faith in MATLAB; their critical thinking is in
abeyance. Consequently, they conclude that the matrix inverse is
X.
Now I direct them to check to see if the
matrix inverse property is satisfied. Now MATLAB becomes the ally
(to the instructor):
Something is clearly wrong! (Make the
students discuss what is wrong and why it is wrong. Identify
why/where things went wrong. Point out the appropriate
interpretation of the matrix returned by rref(aug).) The students
had asked, "Why bother learning extraneous material?" Hopefully,
this lesson has taught them that there is nothing extraneous about
learning concepts. There is, however, something seriously wrong
about blind dependence on computer-generated output.
Other
Examples of Technology Caveats: It is important that students
have an opportunity to see where technology gives false or misleading
results. By seeing examples such as the one in this demo and the
demos in the links below, students can learn to appreciate the
importance of learning mathematical concepts that help them to correctly
interpret (or dismiss) the results from computational
tools.
The following is an evolving list of
interesting examples.
-
What You Don't See. This demo
illustrates that a calculator's display does not always tell the
whole story about the graph of a function.
-
The Trouble with "Exact
Arithmetic." This demo illustrates another example from
linear algebra where matrix computations can be easily
misinterpreted. (under
construction)
-
Existence and Uniqueness Theorem
in Differential Equations: Unimportant?? (tentative
title) This demo illustrates that computer algebra systems can
give multiple symbolic "solutions" to differential equations that
have a unique solution. (under construction)
Credits: This demo was submitted
by
Dr.
David Zitarelli
Mathematics Department
Temple
University
Philadelphia, PA 19122
and is included in Demos with Positive Impact with his
permission.
