A Fast Introduction

inverse matrix 1

What is a matrix ------ (One Matrix many matrices)

Why do they exist

Matrix Terminology • Elements• Rows• Columns

Square Matrix

Adding/Subtracting

Multiplying/ Dividing (Divisions are Multiplications)

The Inverse Matrix (equivalent to 1.0)

Inverse of a Matrix

Solving simultaneous equations

Today’s Goals

—To be able to find the inverse of a Matrix A ~ A’

—To use this for solving simultaneous equations

Recap

The identity Matrix is the equivalent of the number 1 in matrix form

42

21

10

01

43

21

42

21

43

21

10

01

3inverse matrix

Definition of Inverse

The inverse is Defined as AA-1 = A-1A = I

00

01

43

21

??

??

00

01

??

??

43

21

We will find out how to calculate the inverse for 2x2 matrixBut first why is it important ?

Because it will allow us to solve equations of the form

2

1

2

1

2221

1211 .b

b

x

x

aa

aa We will only consider 2x2 Matrix systems That means simultaneous equations

4inverse matrix

Why will it help us solve equations?

Because if we can express a system of equations in the form

bAx Then we can multiply both sides by the inverse matrix

bAAxA 11 And we can then know the values of X because IAA 1

bAx 15inverse matrix

6

Recap Multiplication

106

234

6

1

2

=

-6

23=8+ 3+ 12

-12+0+ 6

=

31

2161

3

5 72

=

12+ 9

10+ 21

= 12344 2 =

24+10 4+8

3

22+ 6 8

6 15 4

inverse matrix

7

Shapes and sizes

• Dimensions and compatibility given by the domino rule

23 31 21

Note matrix multiplication is not commutative. If A is a 3x1 and B is a 1x3 then AB is 3x3 BA is 1x1

To multiply the columns of the first must be equal to

the rows of the second

The dimensions of the result are given

by the 2 outer numbers

inverse matrix

inverse matrix 8

The multiplicative inverse of a matrix

• This can only be done with SQUARE matrices• By hand we will only do this for a 2x2 matrix• Inverses of larger square matrices can be calculated

but can be quite time expensive for large matrices, computers are generally used

Ex A = then A-1 = as AxA-1 = I

31

84

125.0

275.0

10

01

3275.075.0

8823

125.0

275.0

31

84

Finding the Inverse of a 2x2 matrix

inverse matrix 9

dc

ba

Step-1 First find what is called the Determinant

This is calculated as ad-bc

Step-2 Then swap the elements in the leading diagonal

ac

bd

Step-3 Then negate the other elements

ac

bd

ac

bd

cbad

1

Step-4 Then multiply the Matrix by 1/determinant

Example Find Inverse of A

inverse matrix 10

31

84A Determinant (ad-cb) = 4x3-8x1 = 4

Step 1 – Calc Determinant

Step 2 – Swap Elements on leading diagonal

41

83step2

41

83step3

Step 3 – negate the other elements

Step 4 – multiply by 1/determinant

41

83

4

1step4

125.0

275.01A

10

01

3275.075.0

8823

125.0

275.0

31

841AA

check

inverse matrix 11

Find the inverses and check them

10

22C

51

62A

21

205B

10

15.0

20

21

2

1C 1

5.025.0

5.125.1

21

65

4

1A 1

5.01.0

22.0

51

202

10

11B

inverse matrix 12

More inverses to find and check

12

23C

10

01F

01

28D

41

82E

31

42A

21

105B

32

21

32

21

1

1C 1

invertingself10

01

10

01

1

1F 1

45.0

10

81

20

2

1D 1

088)81(42det1 asfoundbecannotE

15.0

25.1

21

43

2

1A 1

25.005.0

5.01.0

51

102

20

1B 1

inverse matrix 13

Applications of matrices

• Because matrices are clever storage systems for numbers there are a large and diverse number of ways we can apply them.

• Matrices are used in to solve equations on computers

- solving equations

• They are used in computer games and multi-media devices to move and change objects in space

- transformation geometry

• We only consider solving equations on Maths1 with using 2x2 matrices

inverse matrix 14

Solving simultaneous equations

• We can use our 2x2 matrices to express 2 simultaneous equations (2 equations about the same 2 variables)

• First we must put them in the correct format • for the variables x & y the format should be

ax + by = mcx + dy = n {where a,b,c,d,m & n are

constants}

ExamplePeter and Jane spend £240 altogether and Peter spends 3

times as much as Jane.let p: what Peter spends and j: what Jane spends

then p + j = 240 (right format a and b = 1 m = 240) p=3j (wrong format)rewrite p-3j = 0 (right format c = 1 d = -3 n = 0)

inverse matrix 15

Solving simultaneous equationsWe can use our 2x2 matrices to express these

simultaneous equations

0

240

31

11

2

1

x

x

constants from the

right hand side

constants from the left hand

side

(the coefficients of p and j)

03

240

yx

yxBecomes in matrix form

UNKNOWNSX ~ x1Y ~ x2

inverse matrix 16

To solve this using the matrix we must get rid of it by using its inverse!

First find the inverse

now use it on both sides of the equation

So Answer is p = 180 j = 60

0

240

31

11

2

1

x

x

25.025.0

25.075.0

11

13

4

1

31

111

0

240

25.025.0

25.075.0

31

11

25.025.0

25.075.0

2

1

x

x

0

240

25.025.0

25.075.0

10

01

2

1

x

x

60

180

2

1

x

x

Format is Ax=B

inverse matrix 17

Summary of method

n

m

y

x

dc

ba

ac

bd

bcad

1

dc

ba1

n

m

ac

bd

bcad

1

y

x

inverse matrix 18

Your Turn solve the following

3x +4y = 55x = 7-6y

x+7y = 1.243y -x = 0.76

8x = 3y -1x+y =-7

7

1

y

x

11

38

76.0

24.1

y

x

31

71

7

5

y

x

65

43

2

1

4

2

2

1

2125

2830

2

1

7

5

35

46

2

1

y

x

Answer x = -1 y = 2

5

2

55

22

11

1

561

211

11

1

7

1

81

31

11

1

y

x

2.0

16.0

2

6.1

10

1

76.024.1

32.572.3

10

1

76.0

24.1

11

73

10

1

y

x

Answer x = -0.16 y = 0.2

Answer x = -2 y = -5