Slide 9.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 9.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Matrix Algebra

Learn the definition of equality of matrices.Learn the definition of matrix addition and scalar multiplication.Learn the definition of matrix multiplication.Learn an application of matrix multiplication to computer graphics.

SECTION 9.2

1

2

3

4


EQUALITY OF MATRICES

Two matrices A = [aij] and B = [bij] are said to be equal, written A = B, if

1. A and B have the same order m n (that is, A and B have the same number m of rows and the same number n of columns.)

2. aij = bij for all i and j. (The (i, j)th entry of A is equal to the corresponding (i, j)th entry of B.)


MATRIX ADDITION

A B aij bij ,

If A =[aij] and B = [bij] are two m n matrices, their sum A + B is the matrix defined by the m n matrix defined by

for all i and j.


SCALAR MULTIPLICATION

cA caij .

If A = [aij] be an m n matrices, and let c be a real number. Then the scalar product of A and c is denoted by cA and is defined by


MATRIX SUBTRACTION

A B A 1 B.

If A and B are two m n matrices, their difference is defined by

Subtraction A – B is performed by subtracting the corresponding entries of B from A.


MATRIX ADDITION AND SCALAR MULTIPLICATION PROPERTIES

Let A, B, and C be m n matrices and c and d be scalars.

1. A + B = B + A2. A + (B + C) = (A + B) + C3. A + 0 = 0 + A = A4. A + (–A) = (–A) + A = 05. (cd)A = c(dA)6. 1A = A7. c(A + B) = cA + cB8. (c + d)A = cA + dA


EXAMPLE 4 Solving a Matrix Equation

Solve the matrix equation 3A + 2X = 4B for X,

A 2 0

4 6

and B

1 3

5 2

.

Solution

3A 2X 4B

2X 4B 3A

X 1

24B 3A

where

Substitute for A and B.


EXAMPLE 4 Solving a Matrix Equation

Solution continued

1

24

1 3

5 2

3

2 0

4 6

1

2

4 12

20 8

6 0

12 18

1

2

2 12

8 10

1 6

4 5

You should check that the matrix X satisfies the given matrix equation.


RULE FOR DEFINING THE PRODUCT ABIn order to define the product AB of two matrices A and B, the number of columns of A must be equal to the number of rows of B. If A is an m p matrix and B is a p n matrix, then the product AB is an m n matrix.


PRODUCT OF 1 x n AND n x 1 MATRICES

Suppose

is a 1 n row matrix and

is a n 1 column matrix. We define the product

A a1 a2 a3 L an

B

b1

b2

M

bn

AB a1b1 a2b2 ... anbn .


MATRIX MULTIPLICATION

Let A = [aij] be an m p matrix and B = [bij] be a p n matrix. Then the product AB is the m n matrix C = [cij], where the entry cij of C is obtained by multiplying the ith row (matrix) of A by the jth column (matrix) of B. The definition of the product AB says that

cij ai1b1 j ai2b2 j ... aipbpj .


EXAMPLE 8 Finding the Product of Two Matrices

Find the products AB and BA, where

A 1 2

1 3

and B

3 2 1

1 2 3

.

Solution

Since A is of order 2 2 and the order of B is 2 3, the product AB is defined and has order 2 3.If AB = C = [cij], each entry cij of C is obtained by multiplying the ith row of A by the jth column of B.



1 2

* *

3 * *

1 * *

1 3 2 1 * *

* * *

Solution continued

C11 is obtained by multiplying the first row of A by the first column of B.

Thus AB 1 2

1 3

3 2 1

1 2 3

AB 1 3 2 1 1 2 2 2 1 1 2 3 1 3 3 1 1 2 3 2 1 1 3 3



Solution continued

The product BA is not defined, because B is of order 2 3 and A is of order 2 2; that is, the number of columns of B is not the same as the number of rows of A.

AB 5 2 7

0 8 8


PROPERTIES OF MATRIX MULTIPLICATION

Let A, B, and C be matrices and let c be a scalar. Assume that each product and sum is defined. Then

1. (AB)C = A(BC)

2. (i) A(B + C) = AB + AC

(ii) (A + B)C = AC + BC

3. c(AB) = (cA)B = A(cB)


EXAMPLE 10 Transforming a Letter

The capital letter L in the figure is determined by six points (or vertices) P1 – P6 . The coordinates of the six points can be stored in a data matrix D, together with instructions stating that these vertices are connected by lines.



P1 P2 P3 P4 P5 P6

0 4 4 1 1 0

0 0 1 1 6 6

D

Given

x-coordinatey-coordinate

Vertex

compute AD. Graph theA 1 0.25

0 1

,

figure corresponding to the matrix AD by connecting the images of the vertices with the



Solution

The columns of the product matrix AD represent the images of the vertices of the letter L.

appropriate lines. This produces the complete transformed image of the original figure.

AD 1 0.25

0 1

0 4 4 1 1 0

0 0 1 1 6 6

10 0.250 14 0.250 14 0.25100 10 04 10 04 11

…



Solution continued

The transformed vertices are plotted on the next slide, along with connecting line segments that correspond to those in the original figure.

11 0.251 11 0.256 10 0.2560111 0116 00 16

…

P1 P2 P3 P4 P5 P6

AD 0 4 4.25 1.25 2.5 1.5

0 0 1 1 6 6



Solution continued

The figure is a result of the transformation represented by the product matrix AD.

Documents

Slide 9.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley