Page 199 - Discrete Mathematics and Its Applications

178 2 / Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

DEFINITION 1 A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called
an m × n matrix. The plural of matrix is matrices. A matrix with the same number of rows
as columns is called square. Two matrices are equal if they have the same number of rows
and the same number of columns and the corresponding entries in every position are equal.

⎡ ⎤
1 1
EXAMPLE 1 The matrix 0 2 ⎦ isa3 × 2 matrix. ▲
⎣
1 3
We now introduce some terminology about matrices. Boldface uppercase letters will be
used to represent matrices.

DEFINITION 2 Let m and n be positive integers and let

⎡ ⎤
a 11 a 12 ... a 1n
⎢a 21 a 22 ... a 2n ⎥
⎢ ⎥
⎢ · · · ⎥
⎥ .
A = ⎢
⎢ · · · ⎥
⎣ ⎦
· · ·
a m1 a m2 ... a mn
The ith row of A is the 1 × n matrix [a i1 ,a i2 ,...,a in ]. The jth column of A is the m × 1
matrix
⎡ ⎤
a 1j
⎢a 2j ⎥
⎢ ⎥
⎥ .
⎢ · ⎥
⎢
⎢ · ⎥
·
⎣ ⎦
a mj
The (i, j)th element or entry of A is the element a ij , that is, the number in the ith row and
jth column of A. A convenient shorthand notation for expressing the matrix A is to write
A =[a ij ], which indicates that A is the matrix with its (i, j)th element equal to a ij .

Matrix Arithmetic

The basic operations of matrix arithmetic will now be discussed, beginning with a definition of
matrix addition.

DEFINITION 3 Let A =[a ij ] and B =[b ij ] be m × n matrices. The sum of A and B, denoted by A + B,is
the m × n matrix that has a ij + b ij as its (i, j)th element. In other words, A + B =[a ij + b ij ].

The sum of two matrices of the same size is obtained by adding elements in the corresponding
positions. Matrices of different sizes cannot be added, because the sum of two matrices is defined
only when both matrices have the same number of rows and the same number of columns.
EXAMPLE 2
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 0 −1 3 4 −1 4 4 −2
We have 2 2 −3 ⎦ + ⎣ 1 −3 0 ⎦ = ⎣ 3 −1 −3 ⎦ . ▲
⎣
3 4 0 −1 1 2 2 5 2