Further Pure 1

Further Pure 1

Lesson 3 –Inverse Matrices

WiltshireReminder from lesson 1

Note that any matrix multiplied by the identity matrix is itself.

And any matrix multiplied by the zero matrix is the zero matrix.

396

813

612

100

010

001

396

813

612

000

000

000

000

000

000

396

813

612

WiltshireInverse Matrix

All operations have an opposite. We discussed in lesson 2 about using matrices to

perform transformations. An inverse matrix will undo the transformation and

return you to where you started. If a matrix is called A, then its inverse is known as A-1. In lesson 1 we briefly met the concept of an identity

matrix (seen on first slide). So if multiplying A by A-1 returns you to where you

started and multiplying by the identity matrix leaves you where you are, we can conclude that AA-1 = A-1A = I

WiltshireChallenge 1

You need to know the general formula for the inverse of a 2 × 2 matrix.

Can you find the inverse to the following matrix? Use the property TT-1 = I

From this we can form 2 pairs of simultaneous equations.

2p + 3q = 1 2r + 3s = 0 p + 4q = 0 r + 4s = 1

10

01

sq

rp

41

32

41

32T


The solutions to these equations are

So the inverse of T is

5

2s

5

3r

5

1q

5

4p

21

34

5

1

5

2

5

15

3

5

41T


Lets now take any 2 × 2 matrix.

Can you use T and T-1 to find the inverse of M and hence the general formula for the inverse of any 2 × 2 matrix?

db

ca

M

ab

cd1

db

ca

21

34

5

1

41

32

1

1

MM

TT

Wiltshire

The Determinant of a 2 × 2 matrix

We have just found the general equation for the inverse of any 2 × 2 matrix.

The Δ symbol is a capital delta and will always be a numerical value.

The value can be calculated from the matrix and is known as the determinant of the matrix.

Using T and T-1 can you spot how to calculate it?

0 for ab

cd1

db

ca 1

MM

21

34

5

1T

41

32T 1

Wiltshire

The Determinant of a 2 × 2 matrix

To calculate the determinant of a matrix M you multiply a by d and subtract b by c.

Below is the official notation.

If the det is zero then the inverse does not exist and the matrix is known as singular.

If the det is not zero then the inverse does exist and the matrix is known as non-singular.

Note: Only square matrices have inverses.

bcaddb

cadet MM


This is above the scope of the course and not required for you to do.

However it is a challenging question that will test your algebraic manipulation skills.

Can you find the inverse of M using the identity below and the method we used a few slides ago.

This will also prove where the formula for the determinant comes from.

10

01

sr

qp

dc

ba

WiltshireQuestions

Find the inverse of the following matrices.

39

722

129

1

229

73

21

34

5

1

41

32

21

54

13

1

41

52

1

1

1

CC

BB

AA

WiltshireInverse of a product

Find the inverse of AB. Lets call the inverse of AB, X. So as we already know X(AB) = I First multiply by B-1 X(AB)B-1 = I×B-1

XA = B-1

Next multiply by A-1 XAA-1 = B-1A-1

X = B-1A-1

This is an important result that you need to know

(AB)-1 = B-1A-1

WiltshireProperties of the determinant

The orange square is an enlargement of the black square by a scale factor 2.

What is the area of the object? Area = 9 units2

What is the area of the image? Area = 36 units2

The transformation performed can be described by the following matrix

What do you notice about the determinant of the matrix and the enlargement shown.

The determinant of a matrix indicates the scale factor of the area of enlargement.

The det T is known as the signed scale factor as it can be negative.

The negative signifies that the rotation direction has been reversed.

20

02T

WiltshireTask

Can you explain how we know that the area of any shape rotated θ degrees anti-clockwise about the origin remains the same.

1θoscθsin

θcossinθ

θsincosθ

22

WiltshireMatrices with det = 0

The determinant of a matrix tells us the scale factor of the areas` enlargement.

What would be the area of a shape transformed by a matrix with det = 0?

The area would be 0. All the points will have been transformed so

what will the image look like? The image will be a straight line. We can see an example of this on the next slide.

WiltshireExample 1

Lets start with a rectangle on a 2D pair of axes.

We can write the co-ordinates of the vertices in matrix form.

Next transform the object using a matrix with a det = 0

The image becomes a series of points that are in a straight line.

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7763

3311

1441

21

21

WiltshireExample 1

7763

7763

3311

1441

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21

In fact although we used a rectangle for the example any point in the plane will transform to the line.

From the diagram its clear to see what the equation of the line will be.

y = x

WiltshireExample 1

`y

`x

y

x

21

21

We can reach the same result as the last slide using an algebraic method.

Lets look at the general co-ordinate (x,y).

Under the transformation we get the co-ordinates (x`,y`)

Using matrix multiplication we can see that.

x + 2y = x`

x + 2y = y` From this we get y` = x + 2y = x` Or y = x

WiltshireExample 2

The plane is transformed by the matrix.

Show that the whole plane is mapped to a straight line and find the equation of this line.

Using matrix multiplication gives us the simultaneous equations.

x` = 2x – y

y` = -4x + 2y From the equations we get y` = -2(2x – y) = -2x` All the points will map to the line y = -2x

23

12T

`y

`x

y

x

24

12

WiltshireExample 2

All points in a plane transform to a straight line. This is because there are infinitely many lines

that transform to a single point.

WiltshireExample 3

For the matrix T find the equation of the line of points that map to (5,-10).

We use matrix multiplication to find what equations will be equal to the co-ordinate (5,-10)

This gives us the equations

2x – y = 5

-4x + y = -10 These two equations give the exact same information.

2x – y = 5

23

12T

10

5

y

x

24

12

WiltshireSummary 1

The inverse of a matrix

Is

Where

0 for ab

cd11

M

db

caM

bcaddb

cadet MM

WiltshireSummary 2

MM-1 = M-1M = I X = B-1A-1

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Further Pure 1