Matrix Inverses via Technology:  Good News, Bad News

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.

1. augment

2. rref

3. 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