2.10: Untitled Page 22
- Page ID
- 18155
Chapter 2
3 x 4 y 2 z 2
8 x 2 y 7 z
5
(2‐38)
5 x 3 y 4 z
3
In matrix form, we express this system of equations according to
3
4
2 x
2
8 2 7
y
5
(2‐39)
5
3
4
z
3
The rule for multiplication implied here is that the first row of the 3 3 matrix multiplies the 3 1 unknown column matrix to obtain the first element of the 3 1
known column matrix. In terms of compact notation, we use Arial font to represent matrices and this leads to the compact form of Eq. 2‐39 given by Au
b
(2‐40)
Here A represents the 3 3 matrix in Eq. 2‐39
3
4
2
A
8 2 7
(2‐41)
5
3
4
while u and b represent the two 3 1 column matrices according to
x
2
u
y
,
b
5
(2‐42)
z
3
The 3 1 matrices are sometimes called column vectors; however, the word vector should be reserved for a quantity that has magnitude and a direction, such as a force, a velocity, or an acceleration. In this text, we will use the phrase column matrix for a n 1 matrix and row matrix for a 1 n matrix. Examples of a 1 4
row matrix and a 4 1 column matrix are given by
6
2
b
3 1 5 6 ,
c
(2‐43)
0
4
31
In a matrix, the numbers are ordered in a rectangular grid of rows and columns, and we indicate the size of an array by the number of rows and columns. The following is a 4 4 matrix denoted by A :
3
1
5
6
4
3
6
2
A
(2‐44)
8
3
2
0
1
5
8
4
Matrices have a well‐defined algebra that we will explore in more detail in subsequent chapters. At this point we will introduce only the operations of scalar multiplication, addition and subtraction. Scalar multiplication of a matrix consists of multiplying each element of the matrix by a scalar, thus if c is any real number, the scalar multiple of Eq. 2‐44 is given by
3 c
1 c
5 c
6 c
4 c 3 c
6 c
2
c
c A
(2‐45)
8 c
3 c
2 c
0
1 c
5 c
8 c
4 c
Matrices have the same size when they have the same number of rows and columns. For example, the two matrices A and B
11
a
12
a
......
1
a n
11
b
12
b
......
1
b n
a
a
...... a
b
b
...... b
21
22
2 n
21
22
2
A
,
n
B
(2‐46)
....
....
......
....
....
....
......
....
a 1
m
m
a 2 ......
m
a n
b 1
m
m
b 2 ......
m
b n
have the same size and the sum of A and B is created by adding the corresponding elements to obtain
11
a
11
b
12
a
12
b
......
1
a n
1
b n
a b
a
b
......
a
b
21
21
22
22
2 n
2 n
A B
(2‐47)
....
....
......
....
a
1
m
b 1
m
m
a 2
m
b 2 ......
mn
a
mn
b
It should be apparent that the sum of two matrices of different size is not defined, and that subtraction is carried out in the obvious manner indicated by