Matrices

Matrices – How to write & What is Order

Apple Orange Pear

Shop 1 1 0.5 0.3

Shop 2 2 1 0.5

5.012

3.05.01

2 rows and 3 columns Matrix has order 2x3(read as “2 by 3”)

Matrices

E.g:

A drink stalls sold 160 cups of coffee, 125 cups of tea and 210 glasses of soft drinks on Monday. On Tuesday, it sold 145 cups of coffee, 130 cups of tea and 275 glasses of soft drinks. On Wednesday, it sold 120 cups of tea, 155 cups of coffee and 325 glasses of soft drinks. Design a matrix to represent this information. State the order of your matrix.

325120155

275130145

210125160

Wed

TueMon

C T SD

325275210

120130125

155145160

SD

TC

M T W

order 3x3

OR

Matrices

Matrix Operations

  

43

21A

11

32B

52

13BA

34

51BA

52

13AB

B – A ?

Observation 1: Matrices must be same order to add or subtract

Matrices

Observation 1: A + B = B + A Addition is commutative

43

21A

11

32B

42

31C

94

44)( CBA ?)( CBA

Observation 2: (A + B) + C = A + (B + C) Addition is associative

(3 × 3) + (4 × 1) = 13 (3 × 2) + (4 × 1) = 10 (1 × 2) + (3 × 1) = 5(1 × 2) + (2 × 1) = 4

Matrices

Scalar Multiplication:

86

422

43

21AA

Matrix Multiplication:

43

21A

11

32B

1310

54BA

Remember: ROW multiply by COLUMN

**Play this slide as slide show to view the multiplication step-by-step

Matrices

An electrical shop sold 5 televisions, 10 VCD players and 15 DVD players on Monday. On Tuesday, it sold 7 televisions, 8 VCD players and 9 DVD players. Given that the price of television is $90, the price of VCD player is $40 and the price of DVD player is $80, find the total sales for Monday and Tuesday.

b

a

80

40

90

987

15105

32 13 12

MT

TV VCD DVDTotal for Monday

Total for Tuesday

$T

$V

$D

Matrices

205080154010905 a

1670809408907 b

b

a

80

40

90

987

15105

Matrices

34

12

21

A

73

12B

642

123C

251

91

138

AB

45345

408BC

4627

91BB

B A Not possible! (2 x 2) (3 x 2)

Not equal! Can only multiply matrices if these numbers are the same!

Matrices

34

12

21

A

73

12B

642

123C

15110247

533421

86682

)( CAB

15110247

533421

86682

)(BCA

Observation 1:

Observation 2:(AB)C = A(BC) Multiplication is associative

AB = BA Multiplication is NOT commutative!

Matrices

The matrix with all entries zero is called a null matrix.

00

00

This matrix is called the identity matrix. The diagonal are all 1’s.

10

01I

00

00

00

00

73

12

73

12A

73

12

10

01

73

12

73

12A

Observation 1: A0 = 0A = 0

Observation 2: AI = IA = A

Practice

Find

(a) BA

(b) BA – 2A

If , find a and b.

341

521A

73

12B

373

32342

b

aAB

Answer

63410

1301BA

0268

3412ABA

a = − 8, b = 238