Matrices

A matrix, A, is a rectangular collection of numbers.

A matrix with “m” rows and “n” columns is said tohave order mxn.

Each entry, or element, in a matrix is denoted byaij, where i stand for the row number and j stands for thecolumn number.

23 13

2 4 1

3 1 2

2 and 1a a

This is an example of a 2 3 matrix

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

This is an example of a square matrix,because the order is nxn

11

21

31

a

a

a

This is an example of a column matrix because it is of the form mx1

11 12 13a a a This is an example of a row matrix because it is of the form 1xn

Two matrices are equal if and only if both matrices are ofthe same order and all the corresponding entries are equal

Adding & Subtracting3 4 5 2 1 2 3 2 4 1 5 2

2 1 3 3 1 6 2 3 1 1 3 6

5 3 7

1 2 3

3 4 5 2 1 2 3 2 4 1 5 2

2 1 3 3 1 6 2 3 1 1 3 6

1 5 3

5 0 9

Scalar Multiple If is a scalar number , is formed by multiplying

element of the matrix by

A each

3 6Let 3 and

3 2

3 6 9 183

3 2 9 6

A

A

1 0 0 1

Let ,2 1 5 2

Find 3 2

A B

A B

1 0 3 0 0 1 0 2 3 3 and 2 2

2 1 6 3 5 2 10 4

3 0 0 2 3 23 2

6 3 10 4 4 1

A B

A B

The transpose of a matrix A, denoted by A’, is formedby swapping the rows for the columns in the matrix

3 73 1 9

1 0 '7 0 12

9 12

A A

It is true in general that for a matrix

the transpose will be of the order

m n

n m

The transpose has the following properties

' ' for any matrix

' ' ' where and are of the same order

' ' for any matrix and any scalar

A A

A B A B A B

A A A

A matrix is said to be symmetric if 'A A A

1 3 5 1 3 5

3 2 1 ' 3 2 1

5 1 7 5 1 7

A A

Note that a symmetric matrix is symmetrical about theleading diagonal

A matrix is said to be skew symmetric if 'A A A

0 3 5 0 3 5

3 0 1 ' 3 0 1

5 1 0 5 1 0

A A

Note that a skew - symmetric matrix must be a square matrix whereall entries in the leading diagonal are equal to zero

Multiplication of MatricesThe matrix product AB can only be found, if the number of columns in matrix A is equal to the number ofrows in matrix B

and

a b g h i

A c d B j k l

e f m n o

and

g h i a b

A j k l B c d

m n o e f

This product is possible since there are three columns in Aand three rows in B

This product is not possible since there are two columns in Aand three rows in B

1 2 2 5 and

3 4 1 3

Find and

A B

AB BA

1 2 2 5 1 2 2 1 1 5 2 3 4 11

3 4 1 3 3 2 4 1 3 5 4 3 10 27AB

2 5 1 2 2 1 5 3 2 2 5 4 17 24

1 3 3 4 1 1 3 3 1 2 3 4 10 14BA

Notice that the matrix products of AB is not equal to BA.This is true in general, but not always. If AB should be equalto BA, the matrices are said to be commute.

2 1 4 and

3 2 2

2 4 1 22 1 4 10

3 4 2 23 2 2 16

A B

AB

Note that BA cannot be found4 3 2 4 5

and 1 2 6 1 2

4 3 2 4 5

1 2 6 1 2

4 2 3 6 4 4 3 1 4 5 3 2

1 2 2 6 1 4 2 1 1 5 2 2

26 19 14

10 2 9

A B

AB

Note, again, that BA cannot be found

2 3 43 2, find , and

0 3M M M M

2 3 2 3 2 9 0 6 6 9 12

0 3 0 3 0 0 0 9 0 9M MM

3 2 9 12 3 2 27 0 18 36 27 54

0 9 0 3 0 0 27 0 27M M M

3 2 3 2 9 12 0 27 36 18 27 54

0 3 0 9 0 27 0 0 27M MM

4 2 2 9 12 9 12 81 0 108 108

0 9 0 9 0 0 81

81 216

0 81

M M M

3 3 2 2It can be shown that M M MM M M In general, , as long as the order

is preserved.

ABC AB C A BC A B C

1 0 01 0

is a 2 2 identity matrix, 0 1 0 is a 3 3 identity matrix0 1

0 0 1

The propertie of the identity matrix " " are

and

I

IA A AI A

1 0 1 0 1 0

0 1 0 1 0 1

a b a c b d a bIA

c d a c b d c d

1 0 1 0 0 1

0 1 1 0 0 1

a b a b a b a bAI

c d c d c d c d

2

3

2 1, find the values of and , such that

3 5

hence express the matrix in the form , where and are integers

A p q A pA qI

A xA yI x y

2 2 1 2 1 4 3 2 5 1 7

3 5 3 5 6 15 3 25 21 22A

2 2 1 1 0

3 5 0 1A pA qI p q

1 7 2 1 0

21 22 3 5 0

1 7 2

21 22 3 5

p p q

p p q

p q p

p p q

equate entries

3 21 7

2 1

2 7 1 13

p p

p q

q q

2 7 13A pA qI A I

3 2

2

7 13

7 13

7 7 13 13

49 91 13

36 91

A AA A A I

A A

A I A

A I A

A I

3 36 91

36 and 91

A A I

x y

Note that this method can be extended to find expressions for A4,A5,….

2

2 2 4 1 1 0

4 2 4 , 2 3 4

2 1 5 0 1 2

Show that ,

Hence obtain

A B

AB kI k

A B

2 2 4 1 1 0 2 4 0 2 6 4 0 8 8

4 2 4 2 3 4 4 4 0 4 6 4 0 8 8

2 1 5 0 1 2 2 2 0 2 3 5 0 4 10

6 0 0 1 0 0

0 6 0 6 0 1 0

0 0 6 0 0 1

6AB I

2 6 6 6A B AAB A AB A I AI A

2

2 2 4 12 12 24

6 4 2 4 24 12 24

2 1 5 12 6 30

A B

The identity element of a matrix addition is

0 0 and the inverse of matrix addition is

0 0O A

a b a bA A

c d c d

Also:

the multiplicative identity element for a 2 2 matrix is

1 0

0 1

A A O

deta b

A ad bcc d

a bA

c d

d badj A

c a

If A 0, the inverse is undefined and the matrix is

called

ad bc

singular

If A 0 then is non-singular

because the inverse is defined.

A

This is referred to as the determinantof the matrix A

This is referred to as the adjugateor adjoint of the matrix A

The inverse, if it exists, of a 2 2 matrix

can be found as follows

Let ,det and adja b d b

A A ad bc Ac d c a

1 adj 1

det

d bAA

c aA ad bc

3 9

2 3A

3 9det 9 and adj

2 3A A

1

3 9 11

3 91 9 9 32 3 2 3 2 19

9 9 9 3

A

1 3 18 91 3

3 9 1 03 3 9 32 3 2 1 2 6 3 0 1

29 3 3 9 3

1AA I

The determinant of a 3 3 matrix

det A b

d

c

e f

g h i

ae f

ah i

d fb

g i

d ec

g h

det A aei afh bdi bfg cdh ceg

aei bfg cdh afh bdi ceg

Alternatively

a b c a b c

d e f d e f

g h i g h i

aei bfg cdh gec hfa idb

1

The inverse of 3 3 matrix can be found by using

the formula det

adj AA

A

Using this it is obvious that the inverse

only exists when det 0A

Before finding the inverse ofa 3 x 3 matrix, we should always evaluate thedeterminant first to make sure the matrix is invertible.

To find the inverse, we use EROs, a method introduced in Unit 1

1 1 1 1 0 0

2 3 1 0 1 0

5 2 3 0 0 1

The target is to transform the first matrix into an identity matrix, but to apply all operations to the second matrix as well

2 1

3 1

2

5

R R

R R

1 1 1 1 0 0

0 5 3 2 1 0

0 3 2 5 0 1

2

3 2

5 then

3

R

R R

3 2 15 5 5

19 315 5 5

1 1 1 1 0 0

0 1 0

0 0 1

3 2 15 5 5

1 1 1 1 0 0

0 1 0

0 0 1 19 3 5

3 5R

1 1 0 18 3 5

0 1 0 11 2 3

0 0 1 19 3 5

1 3

32 35

R R

R R

3 2 15 5 5

19 315 5 5

1 1 1 1 0 0

0 1 0

0 0 1

1 0 0 7 1 2

0 1 0 11 2 3

0 0 1 19 3 5

1 2R R