2 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c ° W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 2 INTRODUCTION TO MATRICES 2.1. Linear Equations Example 2.1.1. Consider the two linear equations 3 x +4 y =11 , 5 x +7 y =19 . It is easy to check the two equations are satisfed when x = 1 and y = 2. We can represent these two linear equations in matrix Form as µ 34 57 ¶µ x y = µ 11 19 , where we adopt the convention that µ •• x y = µ 11 and µ x y = µ 19 represent respectively the inFormation 3 x y = 11 and 5 x y = 19. Under this convention, it is easy to see that µ 10 01 x y = µ x y For every x, y R . Next, observe that µ 7 4 53 = µ , This chapter was written at Macquarie University in 1999.
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2–2 W W L Chen and X T Duong : Elementary Mathematics where, under a convention slightly more general to the one used earlier, we have µ 7 4 •• ¶µ 3 5 = µ 1 , µ 7 4 4 7 = µ 0 , µ 53 3 5 = µ 0 , µ 4 7 = µ 1 , representing respectively (7 × 3)+(( 4) × 5) = 1, (7 × 4)+(( 4) × 7)=0,(( 5) × 3)+(3 × 5) = 0 and (( 5) × 4)+(3 × 7) = 1. It now follows on the one hand that µ 7 4 34 57 x y = µ 10 01 x y = µ x y , and on the other hand that µ 7 4 x y = µ 7 4 11 19 = µ 1 2 . The convention mentioned in the example above is simply the rule concerning the multiplication of matrices. The purpose of this chapter is to study the arithmetic in connection with matrices. We shall be concerned primarily with 2 × 2 real matrices. These are arrays of real numbers of the form µ a 11 a 12 a 21 a 22 , consisting of two rows counted from top to bottom, and two columns counted from left to right. An entry a ij thus corresponds to the entry in row i and column j . 2.2. Arithmetic ADDITION AND SUBTRACTION. Suppose that A = µ a 11 a 12 a 21 a 22 and B = µ b 11 b 12 b 21 b 22 are two 2 × 2 matrices. Then A + B = µ a 11 + b 11 a 12 + b 12 a 21 + b 21 a 22 + b 22 and A B = µ a 11 b 11 a 12 b 12 a 21 b 21 a 22 b 22 .
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2 - ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L...

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