Amatrixis simply a rectangular array of numbers considered as one mathematical object.
When there aremrowsandncolumnsin the array, we have anm-by-nmatrix (written as m×n). We usually denote a matrix with bold capital letters such asA,B, and so on. In general, anm×nmatrix is of the form
A=
⎛
⎜⎜
⎝
a11 a12 . . . a1n
a21 a22 . . . a2n ... ... ... am1 am2 . . . amn
⎞
⎟⎟
⎠ (1)
In this book any array like (1) will be enclosed with large parentheses surrounding the numbers. Note, however, that some writers replace the parentheses in (1) with large square brackets.
The matrixAin (1) is said to haveorderm×n. Themnnumbers that constituteAare called itselementsorentries. In particular,aij denotes the element in theith row and the jth column. For brevity, them×nmatrix in (1) is often expressed as(aij)m×n, or more simply as(aij), if the orderm×nis either obvious or unimportant.
A matrix with either only one row or only one column is called avector. It is usual to distinguish between arow vector, which has only one row, and a column vector, which has only one column. It is usual to denote row or column vectors by small bold letters like xoryrather than capital letters.
E X A M P L E 1 A= 3 −2
5 8
, B=(−1, 2, √
3, 16) , C=
⎛
⎜⎜
⎝
−1 2 8 5 7 6 1 1
⎞
⎟⎟
⎠
are matrices. Of these,Ais 2×2,Bis 1×4 (and so a row vector), andCis 4×2. Also a21=5 andc32=6. Note thatc23is undefined becauseConly has two columns.
E X A M P L E 2 Construct the 4×3 matrixA=(aij)4×3withaij =2i−j.
Solution: The matrixA has 4ã3 = 12 entries. Becauseaij = 2i−j, it follows that a11=2ã1−1=1,a12=2ã1−2=0,a13=2ã1−3= −1, and so on. The complete matrix is
A=
⎛
⎜⎜
⎝
2ã1−1 2ã1−2 2ã1−3 2ã2−1 2ã2−2 2ã2−3 2ã3−1 2ã3−2 2ã3−3 2ã4−1 2ã4−2 2ã4−3
⎞
⎟⎟
⎠=
⎛
⎜⎜
⎝
1 0 −1
3 2 1
5 4 3
7 6 5
⎞
⎟⎟
⎠
Ifm=n, so that the matrix has the same number of columns as rows, it is called asquare matrixof ordern. IfA=(aij)n×n, then the elementsa11, a22, . . . , annconstitute themain diagonalthat runs from the top left (a11) to the bottom right (ann). For instance, the matrix Ain Example 1 is a square matrix of order 2, whose main diagonal consists of the numbers 3 and 8. Note that only a square matrix can have such a main diagonal.
S E C T I O N 1 5 . 2 / M A T R I C E S A N D M A T R I X O P E R A T I O N S 549
E X A M P L E 3 Consider the general linear system
a11x1 + a12x2 + ã ã ã + a1nxn = b1
a21x1 + a22x2 + ã ã ã + a2nxn = b2 . . . . am1x1 + am2x2 + ã ã ã + amnxn = bm
(2)
ofmequations in the unknown variablesxj(j =1,2, . . . , n). It is natural to represent the coefficients of thenunknowns in (2) by them×nmatrixAthat is arranged as in (1). Then Ais called thecoefficient matrixof (2). For instance, the coefficient matrix of
3x1−2x2+6x3 = 5
5x1+ x2+2x3 = −2 is 3 −2 6
5 1 2
One can also represent the numbersbi(i=1,2, . . . , m) on the right-hand side of (2) by an m×1 matrix, or column vector, often denoted byb.
E X A M P L E 4 Consider a chain of stores with four outlets labelled B1, B2, B3, andB4, each selling eight different commodities,V1, V2, . . . , V8. Letaijdenote the dollar value of the sales of commodityViat outletBjduring a certain month. A suitable way of recording this data is in the 8×4 matrix or “spreadsheet”
A=
⎛
⎜⎜
⎝
a11 a12 a13 a14
a21 a22 a23 a24 ... ... ... ... a81 a82 a83 a84
⎞
⎟⎟
⎠
The 8 rows refer to the 8 commodities, whereas the 4 columns refer to the 4 outlets. For instance, ifa73=225, this means that the sales of commodity 7 at outlet 3 were worth $225 for the month in question.
Matrix Operations
So far matrices have been regarded as just rectangular arrays of numbers that can be useful for storing information. The real motivation for introducing matrices, however, is that there are useful rules for manipulating them that correspond (to some extent) with the familiar rules of ordinary algebra.
First, let us agree how to define equality between matrices of the same order. IfA = (aij)m×nandB=(bij)m×nare bothm×nmatrices, thenAandBare said to beequal, and we writeA=B, provided thataij=bijfor alli=1,2, . . . , m, andj =1,2, . . . , n. Thus, two matricesAandBare equal if they have the same orderandif all their corresponding entries are equal. IfAandBarenotequal, then we writeA=B.
E X A M P L E 5 When is 3 t−1
2t u
= t 2v u+1 t+w
?
Solution: Both are 2×2 matrices. Since both have four elements, equality requires the four equations 3 = t, t −1 = 2v, 2t = u+1, and u = t +w to be satisfied. By solving these simultaneous equations, it follows that the two matrices are equal if and only ift =3, v=1, u=5, andw=2. Then both matrices are equal to 3 2
6 5
.
Let us return to Example 4, where the 8×4 matrixArepresents the dollar values of total sales of the 8 commodities at the 4 outlets in a certain month. Suppose that the dollar values of sales for the next month are given by a corresponding 8×4 matrixB=(bij)8×4. The total sales revenues from each commodity in each of the outlets in the course of these 2 months combined would then be given by a new 8×4 matrixC=(cij)8×4, wherecij =aij+bij fori =1, . . . ,8 and forj =1, . . . ,4. MatrixCis called the “sum” ofAandB, and we writeC=A+B.
In general, ifA=(aij)m×nandB=(bij)m×nare two matrices of the same order, we define thesumofAandBas them×nmatrix(aij+bij)m×n. Thus,
A+B=(aij)m×n+(bij)m×n=(aij+bij)m×n (3) So we add two matrices of the same order by adding their corresponding entries.
Ifαis a real number, we defineαAby
αA=α(aij)m×n=(αaij)m×n (4) Thus, to multiply a matrix by a scalar, multiplyeach entry in the matrix by that scalar.
Returning to the chain of stores, the matrix equationB = 2A would mean that all the entries inBare twice the corresponding elements inA—that is, the sales revenue for each commodity in each of the outlets has exactly doubled from one month to the next. (Of course, this is a rather unlikely event.) Equivalently, 2A=A+A.
E X A M P L E 6 ComputeA+B, 3A, and
−12
B, ifA= 1 2 0 4 −3 −1
and B= 0 1 2
1 0 2
. Solution:
A+B= 1 3 2 5 −3 1
, 3A= 3 6 0
12 −9 −3
,
−12
B= 0 −12 −1
−12 0 −1
The matrix(−1)Ais usually denoted by−A, and the difference between the two matrices Aand Bof the same dimension, A−B, means the same asA+(−1)B. In our chain store example,B−Adenotes the (net) change in sales revenue for each commodity from each outlet between one month and the next. Positive components represent increases and negative components represent decreases.
With the definitions given earlier, it is easy to derive some useful rules. LetA,B, andC be arbitrarym×nmatrices, and letαandβbe real numbers. Also, let0denote them×n matrix consisting only of zeros, called thezero matrix. Then:
S E C T I O N 1 5 . 3 / M A T R I X M U L T I P L I C A T I O N 551 R U L E S F O R M A T R I X A D D I T I O N A N D M U L T I P L I C A T I O N B Y S C A L A R S
(a) (A+B)+C=A+(B+C) (b) A+B=B+A
(c) A+0=A (d) A+(−A)=0 (e) (α+β)A=αA+βA (f) α(A+B)=αA+αB
(5)
Each of these rules follows directly from the definitions and the corresponding rules for ordinary numbers.
Because of rule (5)(a), there is no need to put parentheses in expressions likeA+B+C.
Note also that definitions (3) and (4) imply thatA+A+Ais equal to 3A.
P R O B L E M S F O R S E C T I O N 1 5 . 2
1. Construct the matrixA=(aij)3×3, whereaii =1 fori=1,2,3, andaij =0 fori=j. 2. EvaluateA+Band 3Awhen A= 0 1
2 3
and B= 1 −1
5 2
.
3. For what values ofuandvdoes
(1−u)2 v2 3
v 2u 5
6 u −1
=
4 4 u v −3v u−v 6 v+5 −1
? 4. EvaluateA+B, A−B, and 5A−3Bwhen
A= 0 1 −1
2 3 7
and B= 1 −1 5
0 1 9