•Can Conveniently Organize and Store Data • Used to Manipulate

8/14/2019 •Can Conveniently Organize and Store Data • Used to Manipulate

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•Can convenientlyorganize and store data

2 4 5

• Used to manipulatedata



Inventory (July 1)

Small Large

Sales (July)

Small Large

Deliveries (July)

Small Large

Jimmy Johnson 5 4 7 3 6 83 rows 2 columns

Each number in a matrix

Ryan Newman 7 3 2 2 3 4

The dimensions of this matrix are

is called an entry The address of row 2

We can represent our inventory data in a matrix.

small lar e

3 x 2 “3 by 2”

and column 1 is 21m

5 4 Jimmy Johnson1 1 1 2m m

2

7 3

Kyle Petty

Ryan Newman

2 1 2 2m m

m m



Inventory (July 1)

Small Large

Sales (July)

Small Large

Deliveries (July)

Small Large

Jimmy Johnson 5 4 7 3 6 8


Write down the sales matrix S.

12What is =

=

S

S7 3

3

1

32S 2 2 2



Two matrices are equal if theyhave the same dimensions and ifcorresponding entries are

2 4 5 12 x z

equ va ent.

2 3 5 2 5 3 y y

4 x

1 y



We can find the sum (or difference) ofdi i ma r ces w e same dimensions, y

finding the sum or difference ofcorresponding entries.

7 3 6 8

2 2 3 4 13 11

1 5

5 6

1 2



Inventory (July 1)

Small Large

Sales (July)

Small Large

Deliveries (July)

Small Large

Jimmy Johnson 5 4 7 3 6 8


5 4

3 2

7 3

0 1S

6 8

0 0 D

7 3 2 2 3 4

4 9

3 1

8 5



7 3

2 3 1

2 2

3 8 2

A B



We multiply any matrix by a real number k,

y mu p y ng eac e emen n e ma r x yk.

0 1 A Scalar multiplication

2 2

4 A 0 4

8 8



4.1 Using Matrices to Represent Data

Rules and Properties

r r r xA, B , C are matrices with dimensions m n :

Associative (A + B ) + C = A + (B + C )

Additive Identity m n matrix with 0 as all ofits entries

Additive Inverse matrix whose entries areopposites of those in a



m ri Tr n f rm i nRepresent quadrilateralABCD as matrix P

'

' A

3 3 3 2P

( 3,3) A

(3,2) B

'' A''

A B

''C '' D

(3, 2)C

, 2 P

6 4 4 2

'

' D C

1P

3 3 3 12 2 2

2 3 11 1

2 2



Pg 231 #1, 5,6

-



4.24.2 Matrix Multiplication

Multiplication



The Browns scored 4 TD, 1 extra point, 1

, .

The Seahawks scored 3 TD, 3 extrapo n s, an .

6 TD XP 2PT FG

4 1 1 2 1

(4)(6)+ (1)(1)+ (1)(2) +(2)(3)

(3)(6)+ (3)(1)+(0)(2)+(3)(3)

33

30



6

4 1 1 2 1 33

3 uter mens ons

=

Inner dimensions

To multiply two matrices,the inner dimensions must

The outer dimensions become the dimensions

be the same. of the resulting matrix.



Matrix Multiplication

If matrix A has dimensions m n ,

m r .



Matrix Multiplication

o find the entry in row i and column j of AB :

corresponding entries in row i of A and

.



1 2 1 3 5 4



4 7 1 0

3

4 2 1 3 5

4 10 14





Pg 229-230

-







The Identity Matrix for Multiplication

.

Let I be a matrix with the same dimensions and with.

Then AI = IA = A

1



The Inverse of a Matrix

Let A be a square matrix with a dimension ofn x n. If there is an n x n matrix B such thatAB = I, then A and B are inverses of one

another. The inverse of matrix A is denotedby 1

A



3 5 3 2



Determinant of a 2 x 2 Matrix

a b A The determinant ofc d

,

= = A

Matrix A has an inverse if and onl

if det(A) 0



.

Solvin S stems WithMatrix Operations



4 3 55.4 x y

5 2 x 61.2

4 3

A

X

y

55.4

B

CoefficientMatrix

VariableMatrix

ConstantMatrix

Matrix equation



Real Numbers Matrices

ax b AX B1 1

( ) ( )ax ba a

1 1

1 b 1a a 1

X A B

x



5 2 61.2 x

4 3 55.4 y

AX B

1 X A B

10.4 x

10.4 X 4.6 y

.



6 8 18 x y

3 4 3 x 6 8 18 y

AX B1

X A B

"no solution".

Lines are parallel (inconsistent)



4 3 21 x y z

5 2 25 x y z

3 2 2 x y z

5 2 1 25 y

3 1 2 2 z 3

X B

3 X



2 6 x y

2 11 y z

2 16 x z

0 1 2 11 y

2 0 1 16 z 4

X B

5 X



Pg 249

-

r e ou ma r ces



.

Usin Matrix RowOperations



2 3 x y

3 1 5 Augmented Matrix –

1 2 3

Consists of the

constant terms in the



Goal: Transform the matrix into columns

This is called reduced row-echelon form

if the matrix represents an independentsystem.



Elementary Row Operations Notation

•

• Multi l all entries in one row

1 2

2 R R by a nonzero number

•another row

2 1 1

1



1 2 2

1 R R

1 2 3 3 1 5

2 23 R R 0 1 2

3 0 3

1 2 1

1 2 2 R R R 0 1 2

1

3 1 5 1 13

1 0 1

0 1 2



2 2 6 x y z Use calculator: store

3 4 1 x y z augmented matrix anduse rref function.

5 8 3 11 x y z 5 13 x z

0z = 0. This equations tells

ou that z can be an real

. .

0 0

y z

z

number. Dependent.

( 5 13, 3.5 9.5, ) z z z



2 6 4 1 x y z Use calculator: store

3 2 4 x y z augmented matrix anduse rref function.

2 3 7 x y z 0 = 1. This equations tells

ou that there is no solution.

Inconsistent



Pg 257

-

– ca c.

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•Can Conveniently Organize and Store Data • Used to Manipulate