A Definition:(Matrix) J What is a matrix? II. I. Key mathematical terms Matrix

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Matrix

I. Key mathematical terms

Terms Symbol Chinese translation

Matrix Column

Row

II. What is a matrix?

J

onathan is trying to run a small business. He is selling hand-made cookies, cakes and pies for a bake sale. He records his sales during a five-day period – Monday through Friday with a table. He also records the unit prices with another table.

Hand-made

cookies cakes pies

Monday 3 5 2

Tuesday 7 2 1

Wednesday 2 1 5

Thursday 5 0 8

Friday 4 7 6

Table1 Table2

Unit prices Hand-made

cookies 4

Cakes 10

pies 8

We can store these data in two different matrices easily:

1

3 5 2

7 2 1

2 1 5

5 0 8

4 7 6

S

 

 

 

 

 

(matrix about goods sold),

4 10

8 P

  

  

  

(matrix about unit prices)

Definition:(Matrix)

A

matrix is a rectangular array of numbers (or symbols, or functions) arranged into rows and columns. For example, the following are all matrices:

1 0

2 1

1 3 A

 

 

  

 

 

, ^cos ^sin

sin cos

B  

 

  

  

  , C ^a ^b

c d

 

  

  ,

2 2 2

1 1 1

a a

D b b

c c

 

 

  

 

 



^{0 1} ^{2 3}



E  ,

0 1 5 F

  

  

  

, ¹

G  0

  

  , ⁰ H  1

  

 

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A matrix with m rows and n columns can be represented by the following:

11 12 1

21 22 2

1 2

[ ]

1

st n

nd n

m n ij m n

th

m m mn

st th

a a a

A a

a a a m

n

 

 

 

 

 

 

We can say “the order of matrix A is m by n” or “A is a matrix of dimension m×n”.

The individual entries in the matrix are referred to as its elements. The entry in ith row and jth column of a matrix is referred to as the (i,j)th element of a matrix M and is usually denoteda_ij.

<Key> The entries can be arranged in a square brackets or a parentheses.

Example1. (1) Matrix A is a __________ matrix. (Describe the order of the matrix.) (2) (2,3)th element of matrix A above is __________.

III. Some special matrices

Terms Symbol Explanation Chinese translation

Square matrix

Row matrix

Column matrix

Zero matrix

Identity matrix

Diagonal matrix

Symmetric matrix

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Example2. (1) Matrix B, C, D are __________ matrices.

(2) Matrix E is a __________ matrix.

(3) Matrix F, G, H are __________ matrices.

IV. Equality of matrices

T

wo matrices A and B are said to be equal if both of the following conditions hold true:

(1) A and B have the same order.

(2) All corresponding elements of A and B are equal.

Then we can say A and B are equal and can be denoted by A=B.

Example3. Find a, b, x, y with the given condition.

2 4 1

3 3 6 2 1

a x

b y

    

    

   

V. Matrix Operations I: matrix addition and scalar multiplication

J

onathan has recorded the number items sold in the three-week period in the following matrices :

1 2 3

3 5 2 3 5 2 1 5 0

7 2 1 7 2 1 2 6 5

, ,

2 1 5 2 1 5 8 7 7

5 0 8 5 0 8 10 5 11

4 7 6 4 7 6 12 1 16

S S S

     

     

  

     

     

First-week Second-week Third-week The total sales of the first two weeks are:

1 2

3 5 2 3 5 2 3 5 2 3 2 5 2 2 2 6 10 4

7 2 1 7 2 1 7 2 1 7 2 2 2 1 2 14 4 2

2

2 1 5 2 1 5 2 1 5 2 2 1 2 5 2 4 2 10

5 0 8 5 0 8 5 0 8 5 2 0 2 8 2 10 0 16

4 7 6 4 7 6 4 7 6 4 2 7 2 6 2 8 14 12

S S

  

         

            

         

        

         

           

        

           

         



 The total sales of these three weeks are:

1 2 3 1 2 3

6 10 4 1 5 0 6 1 10 5 4 0 7 15 4

14 4 2 2 6 5 14 2 4 6 2 5 16 10 7

( ) 4 2 10 8 7 7 4 8 2 7 10 7 12 9 17

10 0 16 10 5 11 10 10 0 5 16 11 20 5 27 8 14 12 12 1 16 8 12 14 1 12 16 20 15 28

S S S S S S

  

      

         

      

           

         

      

        

      





 

These two operations of matrix are matrix addition and scalar multiplication.

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Matrix addition

[ _{ij m n}] , [ _{ij m n}]

A a _ B b _ are matrices which has the same order.

The addition/subtraction of A and B is given by:

adding corresponding entry subtracting correspondin

[ ] ( )

g ent y

[ ] ( r )

ij ij m n

A B a b A B a b



  

  

Properties of matrix addition

For matrices A, B, C has the same order, then the following holds true:

(1) commutativity: A+B=B+A

(2) associativity: (A+B)+C=A+(B+C) (3) existence of additive identity: A+O=A (4) existence of additive inverse: A+(-A)=O

Example4. For the matrices A and B,

5 3

1 2

2 0

A

 

 

  

 

 

,

2 2

5 1

7 10 B

 

 

  

 

 

, find:

(1) A+B and B+A (2) A+(-A)

(3) A-B and B-A

Example5. For the matrices A, B, C, O,

5 3

1 2

2 0

A

 

 

  

 

 

,

5 3

1 2

2 0

A

 

 

  

 

 

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Scalar multiplication

[ _{ij m n}]

A a _ is a matrix and k is a real number.

The scalar multiplication of A is given by:

[ _{ij m n}] [ _{ij m n}] (multiply every entry of the matrix by )

kAk a _  ka _ k

Properties of scalar multiplication

For matrices A, B has the same order and r,s are real numbers. The following holds true:

(1) associativity: (rs)A=r(sA)=s(rA)

(2) distributivity: (r+s)A=rA+sA, r(A+B)=rA+rB (3) 0A=O

(4) 1A=A

2 4 1 1

3 0 , 0 5

A  B  , find: 3(2A B ) 2( A B )

2 1 1 2

1 4 , 0 3

A   B , find matrix X satisfies : 2(XA)3X2B

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VI. Matrix Operations II: matrix multiplication

J

onathan wants to know the total income of three weeks. We’ve already known that the unit prices can be recorded in Table 2 and total salling S1S2S3 can be recorded in the Table 3 below:

Table2

Unit prices Hand-made

cookies 4

Cakes 10

pies 8

Hand-made

cookies cakes pies

Monday 7 15 4

Tuesday 16 10 7

Wednesday 12 9 17

Thursday 20 5 27

Friday 20 15 28

Table 3 (total sailing) The total income of each day can be calculated by the following:

Monday 7 4 15 10 4 8 (7,15, 4) (4,10,8) Tuesday 16 4 10 10 7 8 (16,10, 7) (4,10,8) Wednesday 12 4 9 10 17 8 (12, 9,17) (4,10,8) Thursday 20 4 5 10 27 8 (20, 5, 27) (4,10,8) Friday 20 4 15 10 28 8 (20,15, 28) (4,10,8

      

       )

If we try to represent the tables into matrices we’ll find that the calculation is just like the inner product of rows and columns of two matrices. That is the multiplication of matrices.

1 2 3

7 15 4

4

16 10 7

, 10

12 9 17

8

20 5 27

20 15 28

S S S S P

 

   

   

 

         

 

 

7 15 4 7 4 15 10 4 8 210

4

16 10 7 16 4 10 10 7 8 220

10

12 9 17 12 4 9 10 17 8 274

8

20 5 27 20 4 5 10 27 8 346

20 15 28 20 4 15 10 28 8 454

S P

    

     

           

      

     

                  

          

     

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Matrix multiplication [ _{ij m n}]

A a _ is a m×n matrix andB[b_{ij n p}]_ k is a n×p matrix.

Then we can define the matrix multiplication of A and B:

[ _{ij m p}]

AB C c _ is a m×p matrix and satisfies

1 1 2 2

( ) ( ) ... ( )

ij i j i j in nj

c  a b  a b   a b (inner product of i-th row and j-th column)

<Key> The numbers of columns of A must equal the numbers of rows of B, then we can form the product matrix AB.

3 2

1 4 3

1 4 ,

2 2 1

2 5

A B

  

 

 

    

, find matrix multiplication (1) AB (2)BA

製作者：國立臺灣師範大學附屬高級中學蕭煜修