Matrices

A matrix is a table or array of numbers arranged in rows and columns

The order of a matrix is given by stating its dimensions.

2 4 1

3 1 2A

This is known as a matrix of order 2 × 3 since it has two rows and three columns.

The element of A in the ith row and jth column is denoted aij.

For example a12 = 4, a21 = 3 and a23 = 2.

1

2

2

B

B is a 3 ×1 column matrix

1 3 5 1C

C is a 1 × 4 row matrix

2 0

1 1D

D is a square matrix of order 2

Addition3 4 5 2 1 2 5 5 7

2 1 3 3 1 6 5 2 9

Scalar Multiplication2 1 6 3

33 5 9 15

Subtraction1 5 3 1 2 6

2 3 3 4 5 1

The Transpose of a matrix

It is sometimes convenient to switch rows and columns. When the rows and columns of matrix A are interchanged, the resulting matrix is called the transpose of A denoted A’ or AT

3 23 4 5

' 4 12 1 3

5 3

B B

2

4 2 4 1

1

T

A matrix is symmetrical if A = AT

1 3 5

3 2 1

5 1 7

A matrix is Skew Symmetric if AT = -A

0 3 5

3 0 1

5 1 0

Note there can only be zeros in the leading diagonal.

Some other Rules:

A B B A ( ') 'A A

( ) ' ' 'A B A B

Page 4 Exercise 1 Questions 1, 2, 3a, 4a, c, e, 6g, i, p, r, t, 7a, f, 9, 10Page 7 Exercise 2

TJ Exercise 1, 2, 3 and 4

Matrix Multiplication

Matrix A can only be multiplied by matrix B when the number of columns in matrix A is the same as the number of rows in matrix B.

A and B might be compatible to form AB but not BA.

The product of an m × n matrix with an n × p matrix will result in an m × p matrix.

r sa b c

t ud e f

v w

ar bt cv as bu cw

dr et fv ds eu fw

8 6 2 4 5

1 3 7 1 2

8 2 6 7 8 4 6 1 8 5 6 2

1 3 7 1 4 3 1 1 5 3 2

58 38 28

19 1 11

Page 10 Exercise 3 Questions 1a, c, 2a, c, k, m, o, 3a, 4, 5a, c Page 11 Exercise 4A Questions 6, 7, 8

TJ Exercise 5

Summary

( ) ' ' '

( ) ( )

( )

AB BA

AB B A

A BC AB C ABC

A B C AB AC

2 2 3 3

The identity matrix is denoted .

1 0 01 0

0 1 00 1

0 0 1

I

I I

Multiplying by the identity matrix does not change the matrix. (i.e.×1)

0.8 0.6A matrix is orthogonal if ' . Prove is orthogonal

0.6 0.8A A I

0.8 0.6'

0.6 0.8A

0.8 0.8 0.6 0.6 0.8 0.6 0.6 0.8'

0.6 0.8 0.8 0.6 0.6 0.6 0.8 0.8A A

1 0

0 1

' hence is orthogonal.A A I A

Page 13 Exercise 4B – as many as you can.

The Determinant of a 2×2 Matrix

Consider a b x e

c d y f

The augmented matrix will be

a b e

c d f

Performing ERO’s we can reduce this to

1

0

b ea a

ad bc af cea a

Hence ,de bf af ce

x yad bc ad bc

A solution exists only if ad – bc ≠ 0

Cayley called this number, ad – bc, the determinant of the matrix. The determinant is denoted by det(A) or |A|.

, det( )a b a b

A A A ad bcc d c d

The Determinant of a 3×3 Matrix

Using the same principals from the previous page on a 3×3 matrix, which you follow on pages 22 and 23, the determinant of a 3×3 matrix is;

det( )

a b ce f d f d e

A d e f a b ch i g i g h

g h i

Page 16 Exercise 5 Questions 1b, d, hPage 25 Exercise 7 Questions 4, 5a, b

The inverse of a 2×2 Matrix

-1The inverse of a 2 2 matrx is denoted A A

1 1, . 0

a b d bA A ad bc

c d ad bc c a

is called the adjoint or adjugate of and is denoted ( )

d bA adj A

c a

If 0, the inverse is undefined. The matrix A is then called singular.ad bc

If 0, the matrix is non singular and invertable.ad bc

12 4(a) If , find .

1 3A A

1 3 416 4 1 2

A

3 221 12

2 3 1(b) Use matrix multiplication to solve

3

x y

x y

2 3 1

1 1 3

x

y

2 3 -1 -31The inverse of is -

1 -1 5 -1 2

1 3 2 3 1 3 11 15 1 2 1 1 5 1 2 3

x

y

1 0 2

0 1 1

x

y

Hence the solution is x = 2, and y = -1.

Page 19 Exercise 6A Questions 1, 2, 4, 8, 9 (some)

TJ Exercise 8

The Inverse of a 3×3 MatrixFrom Gaussian Elimination we can perform ERO's to solve Ax b

1The same operations that will reduce to will reduce to Ax Ix Ib A b

1In other words, the ERO's that convert to will convert to A I I A

•Place A and I side by side, with A on the left.•Perform ERO’s with a view to reducing it to I.•Perform the same ERO’s on I.•When finished I will represent A-1

1

1 1 1

2 3 1 , find

5 2 3

A A

1 1 1 1 0 0

2 3 1 0 1 0

5 2 3 0 0 1

1 1 1 1 0 0

2 2 1 0 5 3 2 1 0

3 5 1 0 3 2 5 0 1

R R

R R

5 1 2 5 0 2 3 1 0

0 5 3 2 1 0

5 3 3 2 0 0 1 19 3 5

R R

R R

1 2 3 5 0 0 35 5 10

2 3 3 0 5 0 55 10 15

3 0 0 1 19 3 5

R R

R R

R

1 2 3 5 0 0 35 5 10

2 3 3 0 5 0 55 10 15

3 0 0 1 19 3 5

R R

R R

R

1 5 1 0 0 7 1 2

2 ( 5) 0 1 0 11 2 3

0 0 1 19 3 5

R

R

1

7 1 2

11 2 3

19 3 5

A

Page 28 Exercise 8 Question 1, 3

TJ Exercise 9, 10, 11.

Transformation Matrices

In computer animation, an object may be drawn by joining lists of points, defined by their coordinates. These points are then transformed according to a rule in order to make the object move.

In this section we will be studying such transformations – linear transformations.

Consider that, under a transformation the point P(x, y) has an image P’(x’, y’). Then

'. We say that is the matrix associated with the transformation.

'

x a b x a b

y c d y c d

A triangle has vertices O(0,0), A(2,5), B(4,0). Find its image under the transformation with associated matrix 2 1

0 2

2 1 0 2 4

0 2 0 5 0

0 9 8

0 10 0

Hence O’ (0,0), A’ (9,10), B’ (8,0).

Constructing a Transformation Matrixa b

Consider the images of (1,0) and (0,1) under the trasnformation c d

1 0

0 1

a b a b

c d c d

(1,0) has image (a,c) and (0,1) has image (b,d).

To find the transformation matrix we only need consider the images of (1,0) and (0,1).

Find the matrix R associated with a reflection in the line y = -x.

Calculate the coordinates of a typical point (x,y) under this transformation.

(1,0) (0,-1)(0,1) (-1,0)

a = 0, c = -1b = -1 d = 0

0 1

1 0R

0 1( )

1 0

x yb

y x

Thus P’ (-y, -x)

(a) Find the matrix k associated with an anticlockwise rotation of 0 about the origin.

(b) Find the coordinates of the image of P(2,4) under this transformation with =600 .

00

(1,0) (cos ,sin )

(0,1) ( sin ,cos )

cos sin

sin cosk

0 0

0 0

2cos60 sin 60( )

4sin 60 cos60b

312 2

3 12 2

2 4

2 4

1 2 3

3 2

Thus '(1 2 3, 3 2)P

Page 32 Exercise 9A Questions 1, 2, 5(some), 6

TJ Exercise 12

END OF TOPIC