|Using a Graphics Calculator to Perform Matrix Operations|
The KML students creating the Mathman clues have the advantage of having graphic
calculators. These calculators make certain operations with matrices MUCH
easier to perform than calculating them by hand.
Adding or subtracting matrices is easier and faster to do by hand. All you have to do is add/subtract the corresponding numbers in the matrices and place the result in the same position in the resulting matrix. Scalar multiplication is also simple to do by hand. All you need to do is multiply the one number (the scalar) times each of the numbers in the matrix one at a time, and write the resulting numbers in the same position in the resulting matrix.
Matrix multiplication and finding/using inverses of matrices are much more tricky. Although students are taught how to do this by hand, due to the amount of time and numerous places for making mistakes involved in working through these operations, using some kind of computing device is generally recommended in actual practice.
Since you probably do not have a graphics calculator, and they are expensive, here is a free graphic calculator simulator you can use. Follow the directions on the webpage to install it on your computer. It does everything the regular calculator does except for the portability issue. (It also clears its memory every time the program is exited.) I would suggest loading it onto a laptop for use out in the letterboxing field. If you only have a desktop computer, you'll have to figure out a method that works for you (or see if you can borrow a graphics calculator from a teenager or math teacher).
(You can try to do these by hand. Use a regular calculator to do matrix multiplication (scroll to "matrix multiplication as combination of rows" (page 8) for an easy method) and inverses of matrices (scroll to page 2 for a 2x2 shortcut). Again, this part of matrix mathematics is one of the more time consuming, confusing, and difficult parts of high school mathematics to learn and do without a graphics calculator (but also one of the most applicable).)
If you are using the simulator or have borrowed a TI-83/84 type graphics calculator, use the following tutorial to learn how to work with matrices on this type of calculator.
Exploring Matrix Operations with the TI-83+
First, enter the dimensions of the matrix giving the number of rows first and the number of columns second. Press the ENTER key after entering each number. The matrix array shown below is a 2 by 3 (2 x 3) matrix. (When entering negative values in the calculator, be sure to use the negative key (-) on the bottom row of the calculator and not the subtraction key.) After the last value is entered in matrix [A] (be sure you hit ENTER after typing the last number so it changes on the screen), press the 2nd key followed by the MODE key to QUIT the matrix. To enter values in other matrices, repeat this process using [B], [C], etc.
[3 4 -2
0.5 -1 6.2]
Look at the matrices below and determine the dimension of each matrix. Give the dimension of each. Give the number of rows first and the number of columns second.
[B] 2 x 2 [C] ____ [D] ____ [E] ____ [F] ____
Enter the values shown below in matrices [B] through [F] in the MATRIX EDIT location of the calculator.
To add, subtract, or multiply matrices, go to the home screen of the calculator. (Press the 2nd key followed by the MODE key to return to the home screen if needed. You may need to press the CLEAR key to clear off the home screen.) If you get a “DIM MISMATCH” error with any operation that means the dimensions of the matrices involved are not appropriate for the operation involved. For example, to add two matrices together they must have the exact same dimensions, so one could not add [A] and [B]..
To add matrices [B] and [E], press the MATRIX key (2nd
x-1) and press the
number 2 key for the name of matrix [B].
Press the addition (+) key and then press the MATRIX (2nd
x-1) key again and choose
number 5 for matrix [E]. Press
(Side note: Multiplying [A] by the resulting matrix would
produce [B]. This is how the
students encoded the message in the first place.)