Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

APP. A] VECTORS AND MATRICES 411

Themhorizontal lists of numbers are called therowsofA, and thenvertical lists of numbers itscolumns. Thus
the elementaij, called theij entry, appears in rowiand columnj. We frequently denote such a matrix by simply
writingA=[aij].
A matrix withmrows andncolumns is called anmbynmatrix, writtenm×n. The pair of numbersmand
nis called thesizeof the matrix. Two matricesAandBare equal, writtenA=B, if they have the same size and
if corresponding elements are equal. Thus, the equality of twom×nmatrices is equivalent to a system ofmn
equalities, one for each corresponding pair of elements.
A matrix with only one row is called arow matrixorrow vector, and a matrix with only one column is called
acolumn matrixorcolumn vector. A matrix whose entries are all zero is called azero matrixand will usually be
denoted by 0.

EXAMPLE A.3

(a) The rectangular arrayA=

[

1 − 45

03 − 2

]

isa2×3 matrix. Its rows are[ 1 ,− 4 , 5 ]and[ 0 , 3 ,− 2 ], and its

columns are

[

1

0

]

,

[

− 4

3

]

,

[

5

− 2

]

(b) The 2× 4 zero matrix is the matrix 0=

[

0000

0000

]

(c) Suppose

[

x+y 2 z+t

x−yz−t

]

=

[

37

15

]

Then the four corresponding entries must be equal. That is,

x+y= 3 ,x−y= 1 , 2 z+t= 7 ,z−t= 5

The solution of the system of equations is

x= 2 ,y= 1 ,z= 4 ,t=− 1

A.4Matrix Addition and Scalar Multiplication

LetA=[aij]andB=[bij]be two matrices of the same size, say,m×nmatrices. ThesumofAandB,
writtenA+B, is the matrix obtained by adding corresponding elements fromAandB. The (scalar)product
of the matrixAby a scalark, writtenkA, is the matrix obtained by multiplying each element ofAbyk. These
operations are pictured in Fig. A-1.

Fig. A-1

Observe thatA+BandkAare alsom×nmatrices. We also define

−A=(− 1 )A and A−B=A+(−B)

The matrix−Ais called thenegativeofA. The sum of matrices with different sizes is not defined.