Row Reduction
by Dan Kalman

This Mathwright Web activity allows students to apply a sequence of row operations to a matrix. The student specifies each row operation descriptively, and observes the results. See the screen shot below.

 

 

Observe that in the pale yellow output window, the row operations and their results are both displayed. To illustrate how the interaction proceeds, here is a description of the three steps that led to the screen above. First, on the part of the screen that looks like this --

-- the student enters 1 in the first yellow box and 2 in the second, and clicks on the word GO! to interchange the first two rows of the matrix. The output window displays the result:

Next, using this input line --

-- the student again fills in the yellow blanks to define a desired operation, entering a -2 in the first blank, a 1 in the second, and a 2 in the third. Clicking the word GO! again produces the next section of the output window.

The next step is to eliminate the first 2 in row 3. This is accomplished similarly, by changing the third yellow box in the input line above to indicate row 3 instead of row 2.

As the student continues to perform row operations, a record is generated in the output window showing each operation performed, and the successive results.

 

The operations that may be invoked are interchanging two rows, multiplying a row by a scalar, and adding a multiple of one row to another. It is possible to begin with any matrix the user cares to enter, or to generate a random matrix. There is also a link which adjoins an identity matrix to the right of the given matrix. The activity opens up in exact fraction mode -- meaning that all operations are performed in exact arithmetic using fractions. However, the user can switch to decimal mode, which employs floating point arithmetic. In this mode, the user also specifies the number of decimal digits to be displayed in the output window.

 

Sample Classroom Activities: Begin with an identity matrix and apply various row operations to observe the creation of elementary row matrices; Execute a row operation, and then find a way to undo that operation and restore the original matrix.

 


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