# 2.10: Untitled Page 22

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## 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