Download Advanced Higher Mathematics Unit 3

Advanced Higher Mathematics Unit 3
Notes on Matrices
1.
Some Definitions
A matrix is a rectangular array of numbers. Each number in the matrix is called an element of the
matrix.
The order of the matrix is described by writing the number of rows followed by the number of
columns.
1
 5 0 9
 2 3




Thus if A  
 , B   3  , C   0 8 4  ,
1
4


3
 3 0 1 
 


then A is a matrix of order 2  2 , B has order 3 1, and C has order 3  3.
More commonly, we say that A is a square matrix of order 2, B is a square matrix of order 3,
and so on.
In this course we deal mostly with square matrices.
The transpose of a matrix A is obtained by interchanging the rows and columns of A .
the transpose of A is written A , or A t .
 1 3
1 0
Thus A  
  A  
.
 0 5
3 5
The zero matrix, O, is the matrix consisting entirely of zeros.
0 0
Thus 
 is the zero square matrix of order 2.
0 0
2.
Addition and Subtraction of Matrices
Matrices of the same order can be added or subtracted in the obvious way.
1 2
3 0 
4 2 
For example, A  
, B  
 AB 
.
3 1
 1 2 
 4 1
3.
Multiplication by a Number
Again, this is defined in the obvious way.
3 6
 6 0 
With A and B as above, we have 3A  
 and 2B  
.
9 3
 2 4 
4.
Multiplication of Matrices
First, we consider the 2  2 case.
a 
a
b b 
If A   11 12  and B   11 12  , then the matrix product AB is defined by:
 a21 a22 
 b21 b22 
a11b12  a12b22 
a b a b
AB   11 11 12 21
.
 a21b11  a22b21 a21b12  a22b22 
It is useful to remember that the rows of the first matrix are multiplied into the columns of the
second.
Worked examples will be done in class to ensure that you are competent at this topic.
Now we consider the 3  3 case:
 a11 a12 a13 
 b11 b12


If A   a21 a22 a23  and B   b21 b22
a

b b
 31 a32 a33 
 31 32
b13 

b23  , then the matrix product AB is defined by:
b33 
 a11b11  a12b21  a13b31 a11b12  a12b22  a13b32

AB   a21b11  a22b21  a23b31 a21b12  a22b22  a23b32
a b a b a b
 31 11 32 21 33 31 a31b12  a32b22  a33b32
a11b13  a12b23  a13b33 

a21b13  a22b23  a23b33  .
a31b13  a32b23  a33b33 
In general, the matrix product AB is defined provided that the number of columns in matrix A is
equal to the number of rows in matrix B.
If A is an m n matrix and B is an n  p matrix then AB is an m  p matrix.
Examples on this will be done in class, but in this course we deal almost always with square
matrices.
It is important to note that in the world of matrices, AB  BA in general. This will be confirmed
in class.
5.
The Unit Matrix
1 0
a b
 a b  1 0   a b 
Let I  
 and A  
 . Then AI  


.
0 1
c d
 c d  0 1   c d 
 1 0  a b   a b 
Also, IA  


.
 0 1  c d   c d 
Hence we have AI  IA  A, and in the world of 2  2 matrices I has the same properties as 1
has in the world of real numbers under multiplication.
I is said to be the unit matrix of order 2. [sometimes called the identity matrix].
The notation I 2 is often used to denote the unit matrix of order 2.
1 0 0


In the world of 3  3 matrices the unit matrix is I 3   0 1 0  , or just I if there is no ambiguity
0 0 1


about the order.
6.
Some Laws of Matrices
(i)
The Distributive Law
Where matrix multiplication is defined, A(B  C)  AB  AC and ( A  B)C  AC  BC.
(ii)
The Associative Law
Where matrix multiplication is defined, A(BC)  ( AB)C.
Given this fact, we can write ABC without there being any ambiguity.
Numerical examples illustrating these results will be done in class, and then we will prove them.
(iii)
Commutativity
As you already know, in the world of matrices the product AB is not in general equal to
the product BA. [where the two products are defined].
But if A and B are matrices such that AB  BA, then A and B are said to commute.
7.
The Inverse of a Square Matrix
4 7
 2 7 
Let A  
 and B  
 . Then, by calculation, it is easy to verify that AB  BA  I.
1 2
 1 4 
In this case, A is said to be the inverse of B. [and B is the inverse of A ].
The inverse of a matrix A, if it exists, is usually written A 1 .
A 1 , if it exists, is unique. [i.e. there is exactly one inverse matrix for A ]. This will be proved in
class.
If matrix A has an inverse, it is said to be invertible; if it does not have an inverse, it is said to be
singular, or non-invertible.
Finding the Inverse of a Square Matrix of Order 2
a b
1  d b 
Let P  
 and Q 

.
ad  bc  c a 
c d
Then, if ad  bc  0, it is easy to verify that PQ  QP  I.
a b
1  d b 
1
Hence, provided that ad  bc  0, if P  
 , then P 

.
ad  bc  c a 
c d
If ad  bc  0, then P 1 does not exist.
This shows how to calculate the inverse of a 2  2 matrix, if the inverse exists.
An interesting comparison is with the system of whole numbers, under the operation of
multiplication:
Here the “unit” element is the number 1, and the “inverse” of the number 17,say, is 171 , since
7  171  171 17  1.
But for the number zero, there is no number x such that 0  x  x  0  1, so 0 has no “inverse” in
the real numbers under multiplication.
Finding the Inverse of a Square Matrix of Order 3
This will be done in class, using Elementary Row Operations.
Examples on finding the inverse of a matrix will be done in class.
An important theorem on inverses is, where A and B are invertible matrices of the same order,
(AB)1  B1 A1
1
1
1
1
And generalising this result, we get (A1A2 A3A4  An )1  An An1 A2 A1 .
8.
The Determinant of a Matrix
a b
Let A  
 . Then the determinant of matrix A, written A or det( A ), is defined by
c d
A  ad  bc.
 a11 a12
For the 3  3 case, when A   a21 a22
a
 31 a32
A  a11
a22
a23
a32
a33
a13 

a23  , the determinant of matrix A is defined by
a33 
 a12
a21 a23
a31
a33
 a13
a21 a22
a31
a32
.
Numerical examples will be done in class.
Some important ideas involving determinants are:
(i)
(ii)
(iii)
(iv)
A 1 exists  det( A)  0.
det( AB)  det( A) det(B).
det(I )  1.
1
det  A 1  
.
det( A)
These propositions will be studied in class.
9.
Orthogonal Matrices
A matrix A is orthogonal if A is square and AA  AA  I. [i.e. A 1  A. ]
Some propositions on orthogonal matrices are:
(i)
(ii)
A , B orthogonal n  n matrices  AB orthogonal.
A orthogonal  A orthogonal.
Examples on orthogonal matrices will be done in class.
10.
Transformations
In 2-dimensional space, certain transformations can be represented by 2  2 matrices.
These will be dealt with in a separate document.
 cos  sin  
But the one which we should remember is R  
 . This represents a rotation of 
 sin  cos 
radians anticlockwise about the origin.
This transformation, and the others, will be derived in class.
11.
Symmetric Matrices
Matrix A is symmetric if its elements are symmetric about its main diagonal, i.e. if  A ij   A  ji .
Matrix A is skew-symmetric if  A ij    A  ji .
Clearly, if A is skew-symmetric then  A ii    A ii   A ii  0. Hence the elements in the main
diagonal of a skew-symmetric matrix are all zero.
12.
Solving Linear Equations Using Matrices
You already have encountered the method of “Gaussian Elimination” in Unit 1. Another approach
to the solution of a set of linear equations is, provided A 1 exists:
Ax  b
A
1
 Ax   A 1b
A A x  A
1
1
b
Ix  A 1b
x  A 1b
Hence if we have the technology, (T.I.83 in our case!), we can obtain the solution to a set of linear
equations, at least in the case where there is a unique solution.
Recall also that A 1 exists  det A  0, so we have a unique solution provided det A  0.
In the case where det A  0, there is either an infinite number of solutions, or there are no
solutions. These cases, for 3  3 systems, have been covered in Unit 1 and will be revised in class.
13.
Overview
This document has been done to give you a set of basic notes on Matrices. In order to become
proficient at all the techniques outlined here you will have to do the examples from the sheets,
books and old examination papers.