Matrices A matrix is a table or array of numbers arranged in rows and columns The order of a matrix...
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Transcript of Matrices A matrix is a table or array of numbers arranged in rows and columns The order of a matrix...
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