Unit 6 : Matrices

MATRIX: A rectangular arrangement of numbers in rows and columns.

The ORDER of a matrix is the number of the rows and columns.

The ENTRIES are the numbers in the matrix.

502

126rows

columns

This order of this matrix is a 2 x 3.

67237

89511

36402

3410

200

318 0759

20

11

6

0

7

9

3 x 3

3 x 5

2 x 2 4 x 1

1 x 4

(or square matrix)

(Also called a row matrix)

(or square matrix)

(Also called a column matrix)

To add two matrices, they must have the same order. To add, you simply add corresponding entries.

34

03

12

70

43

35

)3(740

0433

13)2(5

44

40

23

9245

3108

2335

2571

)1(8 70 51 23

55 34 32 )2(9 =

= 7 7 4 5

0 7 5 7

To subtract two matrices, they must have the same order. You simply subtract corresponding entries.

232

451

704

831

605

429

2833)2(1

)4(65015

740249

603

1054

325

724

113

810

051

708

342

=

5-2

-4-1 3-8

8-3 0-(-1) -7-1

1-(-4)

2-0

0-7

=

2 -5 -5

5 1 -8

5 3 -7

In matrix algebra, a real number is often called a SCALAR. To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar.

14

024

416

08

)1(4)4(4

)0(4)2(4

86

54

30

212

)8(360

52412

-2

6

-3 3

-2(-3)

-5

-2(6) -2(-5)

-2(3) 6 -6

-12 10

2 5 3 6 7 0

9 7 5

7 9

3 12

7

16 28 4

Multiplication of Matrices

Scalar multiplication – multiply the entire matrix by a number

Example 3:

2 9

3 0 1

5 12

6 27

0 3

15 36

Multiplication of Matrices

Matrix multiplication – two matrices can only be multiplied if the number of columns in the first equals the number of rows in the second. 2x3 could be multiplied with a 3x4

could not multiply 3x4 and 3x4The dimensions of the product matrix (what you get after you multiply) will be the number of rows from the first and the number of column from the second.

When you multiply the 2x3 and the 3x4, the product will be a 2x4

Matrix multiplication – to multiply two matrices, you multiply each row in the first by each column in the second.

Row by column, row by column Multiply them line by line Add the products, form a matrix Now you're doing it just fine

Matrix multiplication Song

Example 4:

3 21 2 0

0 43 5 2

1 1

2x3 and 3x2…can multiply and the product will be a 2x2

Check :

(1)( 3) (2)(0) (0)(1) (1)(2) (2)(4) (0)(1)

(3)( 3) ( 5)(0) (2)(1) (3)(2) ( 5)(4) (2)(1)

3 10

7 12

Example 5:A motor manufacturer, with three separate factories, makes two types of car -one called “standard” and the other called “luxury”. In order to manufacture each type of car, he needs a certain number of units of material and a certain number of units of labour each unit representing £300. A table of data to represent this information could be Type Materials Labour

Standard 12 15

Luxury 16 20

The manufacturer receives an order from another country to supply 400 standard cars and 900 luxury cars.

He distributes the export order as follows:

Location Standard Luxury

Factory A 100 400

Factory B 200 200

Factory C 100 300

Using matrix multiplication, find a matrix to represent the number of units of material and labour needed to complete the order.

Solution:

100 40012 15

200 20016 20

100 300

100 12 400 16 100 15 400 20

200 12 200 16 200 15 200 20

100 12 300 16 100 15 300 20

7600 9500

5600 7000

6000 7500

Determinants

Every square matrix has a number associated with it called a determinant.

Second – order determinant denoted by:

deta b a b

orc d c d

= ad - bc

Product of the diagonal going down minus the product of the diagonal going up

Example 6:

3 10det

4 5Find

Solution:

det A= (3)(-5) – (10)(4)= -15 – 40 = -55

3 10

4 5

Let A =

Example 7:

1 43 0

Find

Solution: Let A = 1 4

3 0

det A= (1)(0) – (-4)(3) = 0 – -12= 12

Identity and Inverse Matrices

Identity matrix is a square matrix that when multiplied by another matrix, the product equals that same matrix.

:

1 0 0 01 0 0

1 0 0 1 0 0, 0 1 0 , ,

0 1 0 0 1 00 0 1

0 0 0 1

I dentity matrix

etc

Identity Matrix has 1 for each element on the main diagonal and 0 everywhere else.

matrix times inverse = identity matrix1A A I

Not every matrix has an inverse.Not every matrix has an inverse.

Requirements to have an Inverse

• The matrix must be square (same number of rows and columns). The determinant of the matrix must not be zero.

• A square matrix that has an inverse is called invertible or non-singular.

• A matrix that does not have an inverse is called singular. The determinant of the matrix equal zero.

Inverse of a second order matrix (2 x 2):a b

c d

1 1det

d bA

c aA

Change the place of a and d and change the signs of c and b.

Example 8:

Find the inverse of 1 2

3 4

Solution:

11 2

3 4

4 21

3 11(4) 3(2)

2 1

1.5 0.5

Solving Simultaneous Equations using inverse matrix

Consider the simultaneous equationsx + 2y = 4

3x − 5y = 1

In Matrix Form : 1 2 4

3 5 1

x

y

Let , and1 2

3 5A

xX

y

4

1B

We have AX = B.This is the matrix form of the simultaneous equations. Here the unknown is the matrix X, Since A and B are already known. A is called the matrix of coefficients.