Greenwood press

60
LOGISTIC FUNCTIONS

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MATRICES
61
MATRICES
A matrix is a rectangular array of numbers. Operations that correspond to
addition, multiplication, and powering of real numbers provide rules for com-
bining matrices. Inverses of matrices correspond to reciprocals of real numbers.
In addition, there are specific operations on some matrices, such as those used in
game theory and graph theory that transform elements of the matrix to point the
way to best decisions.
Matrices are used to solve large systems of simultaneous equations. High
school students usually see matrices as a way to rewrite systems of equations. 
For example, 
5x+3y=7
2x−y=5
can be replaced by 

5
3
2 −1

x
y
=

7
5
. This 
change seems very simple, but it generalizes the system to a matrix system Ax =
b, which has a solution (if it exists) of x = A
–1
b. The problem of accurately com-
puting the inverse A
-1
for large matrices is difficult, even with high-speed com-
puter processors. This remains a critical issue for mathematicians and computer
analysts, because scientists in fields as widely diverse as astronomy, weather
forecasting, statistics, economics, archeology, water management, weapons races
between countries, chicken production, airline travel routes, investment banking,
marketing studies, and medical research rely on the efficient reduction of large
matrices of information.
Matrix multiplication can provide a more secure secret code than simple
replacement ciphers. Replacement ciphers (sometimes called Caesar ciphers in
honor of the Roman emperor Julius Caesar, who used them in his military cam-
paigns) encode a message by replacing each letter with another. The problem
with these ciphers is that certain letters occur more frequently in languages than
do others. If “z” and “m” occur most frequently in an English-language coded
document, it is likely that the most frequent letter is hiding “e” and the next, “t.”
This one-to-one correspondence makes it easy to decode secret messages written
in replacement ciphers. If the code is written with numbers that are encoded with
multiplication by a matrix, the same letter encodes to different letters, depending
on its position in the message. The English-language frequency distribution is
then destroyed, so it is far more difficult for code breakers to decipher the mes-
sage. Recipients who have the encoding matrix, however, can quickly decode the
message by multiplication with the inverse of the matrix.
Some matrices describe transformations of the plane. Common geometric
movements of figures, such as reflections and rotations can be written as 2 
× 2
matrices. The table below shows some common transformation matrices.