9.4 Matrices 569 Five hundred students at the University of Delaware were asked if they were in favor of or opposed to an increase in their student fees to pay for building a new meeting room for student clubs. Students were also asked to indicate whether they were a freshman, sophomore, junior, or senior. How can the responses from the survey be displayed? In this section, we will introduce a method used to display information such as responses from a survey. Matrices SECTION 9.4 LEARNING GOALS Upon completion of this section, you will be able to: 7 Add matrices. 7 Subtract matrices. 7 Multiply matrices by a real number. 7 Multiply two matrices. 7 Show that matrices can be used to form a commutative group. Why This Is Important We often need an efficient way to display information such as responses from a survey, a company’s current inventory, or when creating a budget. So far in this chapter we have discussed various mathematical systems that are also groups. In this section we will introduce mathematical systems that involve matrices. We will also show that certain sets of matrices, along with the operation of addition, form commutative groups. We begin with some basic definitions. Matrices A matrix is a rectangular array of real numbers, symbols, or expressions. An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be used to display information and to solve systems of linear equations. The following matrix displays the responses from the survey of 500 students at the University of Delaware regarding an increase in their student fees. 80. Rolling Wheel The wheel shown is to be rolled. Before the wheel is rolled, it is resting on number 0. The wheel will be rolled at a uniform rate of one complete roll every 4 minutes. In exactly 1 year (not a leap year), what number will be at the bottom of the wheel? 0 0 2 1 3 Research Activities 81. The Enigma Cipher Machine Do research on the Enigma ciphering machine (see the Did You Know? box on page 565). Write a paper on the role the machine played prior to and during World War II. 82. Carl Friedrich Gauss The concept and notation for modular systems were introduced by Carl Friedrich Gauss in 1801. Write a paper on Gauss’s contribution to modular systems. 77. Birthday Question During a certain year, Clarence’s birthday is on Monday, April 18. a) If next year is not a leap year, on what day of the week will Clarence’s birthday fall next year? Tuesday b) If next year is a leap year, on what day of the week will Clarence’s birthday fall next year? Wednesday 78. More Birthday Questions During a certain leap year, Josephine’s birthday is on Tuesday, May 18. What is the minimum number of years in which Josephine’s birthday will fall on a a) Friday? 3 years b) Saturday? 9 years Recreational Mathematics 79. Cryptography Read the Recreational Math box on page 562. Using the same coding circle and a code key of the letter e, decipher the following message. 1619201 17109 1210 514245 21201 1023 Knowledge is power. Kadettmann/123RF
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