# 2.10: Untitled Page 22

## 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 

Units

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

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