Programme 5: Matrices
PROGRAMME 5
MATRICES
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
A matrix is a set of real or complex numbers (called elements) arranged in
rows and columns to form a rectangular array.
A matrix having m rows and n columns is called an m × n matrix.
For example:
5


6
7 2 
3 8 
is a 2 × 3 matrix.
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Row matrix
A row matrix consists of a single row. For example:
4
3 7 2
Column matrix
A column matrix consists of a single column. For example:
6
 
 
 3
 
8
 
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Double suffix notation
Each element of a matrix has its own address denoted by double suffices,
the first indicating the row and the second indicating the column. For
example, the elements of 3 × 4 matrix can be written as:
a
 11

 a21

a
 31
STROUD
a12
a22
a32
a13 a14 
a23 a24 

a33 a34 
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrix notation
Where there is no ambiguity a matrix can be represented by a single general
element in brackets or by a capital letter in bold type.
a
 11

 a21

a
 31
STROUD
a12
a22
a32
a13 a14 
a23 a24  can be denoted by  aij  or by A

a33 a34 
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Equal matrices
Two matrices are equal if corresponding elements throughout are equal.
( ) ( )
A = B that is aij = bij if aij = bij for all values of i and j
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Addition and subtraction of matrices
Two matrices are added (or subtracted) by adding (or subtracting)
corresponding elements. For example:
4


5
STROUD
2 3   1 8 9   4 1 2  8 3  9 


 
 
7 6   3 5 4   5  3 7  5 6  4 
 5 10 12 

 

 8 12 10 
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Multiplication of matrices
Scalar multiplication
Multiplication of two matrices
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Multiplication of matrices
Scalar multiplication
To multiply a matrix by a single number (a scalar), each individual element
of the matrix is multiplied by that number. For example:
3
4 
6
2 5  12 8 20 

 
1 7   24 4 28 
That is:
k  aij    kaij 

STROUD



Worked examples and exercises are in the text
Programme 5: Matrices
Multiplication of matrices
Multiplication of two matrices
Two matrices can only be multiplied when the number of columns in the
first matrix equals the number of rows in the second matrix.
The ijth element of the product matrix is obtained by multiplying each
element in the ith row of the first matrix by the corresponding element in the
jth column of the second matrix and the element products added.
For example:
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Multiplication of matrices
Multiplication of two matrices
æ a
a12
11
If A = ç
çè a21 a22
a13
a23
æ a
a12
11
then A.B = ç
çè a21 a22
STROUD
æ b
11
ö
ç
÷ and B = ç b21
÷ø
ç b
è 31
a13
a23
æ b
ö ç 11
÷ .ç b21
÷ø ç
è b23
ö
÷
÷
÷
ø
ö
÷ æ a11b11 +a12 b21 +a13b31
÷ =ç
÷ çè a21b11 +a22 b21 +a23b31
ø
Worked examples and exercises are in the text
ö
÷
÷ø
Programme 5: Matrices
Multiplication of matrices
Multiplication of two matrices
If A   aij  is an n  m matrix and
B   bij  is an m  q matrix then
C = A.B   cij  is an n  q matrix where
m
c   aik bkj
ij k 1
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Transpose of a matrix
If a new matrix is formed by interchanging rows and columns the new
matrix is called the transpose of the original matrix. For example, if:
 4 6
 4 7 2
A   7 9  then AT  

6
9
5


 2 5


STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Special matrices
Square matrix
Diagonal matrix
Unit matrix
Null matrix
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Special matrices
Square matrix
A square matrix is of order m × m.
A square matrix is symmetric if aij  a ji . For example:
1


2

5

2
8
9
5 
9 

4 
A square matrix is skew-symmetric if aij  a ji . For example:
 0


 2

 5

STROUD
2
0
9
5 
9 

0 
Worked examples and exercises are in the text
Programme 5: Matrices
Special matrices
Diagonal matrix
A diagonal matrix is a square matrix with all elements zero except those on
the leading diagonal. For example:
5


0

0

STROUD
0
2
0
0 
0 

7 
Worked examples and exercises are in the text
Programme 5: Matrices
Special matrices
Unit matrix
A unit matrix is a diagonal matrix with all elements on the leading diagonal
being equal to unity. For example:
1

I   0

0

0
1
0
0 
0 

1 
The product of matrix A and the unit matrix is the matrix A:
A.I = A
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Special matrices
Null matrix
A null matrix is one whose elements are all zero.
0

0   0

0

0
0
0
0 
0 

0 
Notice that
A.0 = 0
But that if A.B = 0 we cannot deduce that A = 0 or B = 0
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Determinant of a square matrix
Singular matrix
Cofactors
Adjoint of a square matrix
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Determinant of a square matrix
Singular matrix
Every square matrix has its associated determinant. For example, the
determinant of
5


0

8

2
6
4
1 
5
2
1
3  is 0

7 
8
6
4
3 150
7
The determinant of a matrix is equal to the determinant of its transpose.
A matrix whose determinant is zero is called a singular matrix.
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Determinant of a square matrix
Cofactors
Each element aij of a square matrix has a minor which is the value of the
determinant obtained from the matrix after eliminating the ith row and jth
column to which the element is common.
The cofactor of element aij is then given as the minor of aij multiplied by
 1i  j
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Determinant of a square matrix
Adjoint of a square matrix
Let square matrix C be constructed from the square matrix A where the
elements of C are the respective cofactors of the elements of A so that if:
A   aij  and Aij is the cofactor of aij then C   Aij 


Then the transpose of C is called the adjoint of A, denoted by adjA.
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Inverse of a square matrix
If each element of the adjoint of a square matrix A is divided by the
determinant of A then the resulting matrix is called the inverse of A,
denoted by A-1.
A1 
1
 adjA 
det A
Note: if det A = 0 then the inverse does not exist
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Inverse of a square matrix
Product of a square matrix and its inverse
The product of a square matrix and its inverse is the unit matrix:
A.A-1 = A-1.A = I
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
The set of n simultaneous linear equations in n unknowns
can be written in matrix form as:
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
Since:
A.x = b then
A1.Ax = A1.b that is
I.x = A1.b and I.x = x
The solution is then:
x = A1.b
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
Gaussian elimination method for solving a set of linear equations
Given:
æ a
11
ç
ç a21
ç
ç
çè an1
a12
a13
a1n
a22
a23
a2n
an2
an3
ann
öæ x
1
֍
÷ ç x2
֍
֍
÷ø çè xn
æ
ö çç b1
÷ ç
÷ çç b2
÷ =ç
÷ çç
÷ø ç b
ç
n
è
ö
÷
÷
÷
÷
÷
÷
÷
÷
÷
÷
ø
Create the augmented matrix B, where:
æ
ç a11
ç a
B = ç 21
ç
ç a
çè n1
STROUD
a12
a13
a1n
a22
a23
a2n
an2
an3
ann
b1 ö
÷
b2 ÷
÷
÷
÷
bn ÷ø
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
Gaussian elimination method for solving a set of linear equations
æ
ç a11
ç a
B = ç 21
ç
ç a
çè n1
a12
a13
a1n
a22
a23
a2n
an2
an3
ann
b1 ö
÷
b2 ÷
÷
÷
÷
bn ÷ø
Eliminate the elements other than a11 from the first column by subtracting
a21/a11 times the first row from the second row, a31/a11 times the first row
from the third row, etc. This gives a new matrix of the form:
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
Gaussian elimination method for solving a set of linear equations
æ
ç a11
ç 0
ç
ç
ç 0
çè
a12
a13
a1n
c22
c23
c2n
cn2
cn3
cnn
b1 ö
÷
d2 ÷
÷
÷
÷
d n ÷ø
This process is repeated to eliminate the ci2 from the third and subsequent
rows until a matrix of the following form is arrived at:
æ a
ç 11
ç 0
ç
ç 0
ç
çè 0
STROUD
a1,n-2
a1,n-1
a1n
pn-3,n-2
pn-2,n-1
pn-2,n
0
pn-1,n-1
pn-1,n
0
0
pnn
b1 ö
÷
q2 ÷
÷
÷
÷
qn ÷ø
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
Gaussian elimination method for solving a set of linear equations
æ a
ç 11
ç 0
ç
ç 0
ç
çè 0
a1,n-2
a1,n-1
a1n
pn-3,n-2
pn-2,n-1
pn-2,n
0
pn-1,n-1
pn-1,n
0
0
pnn
b1 ö
÷
q2 ÷
÷
÷
÷
qn ÷ø
From this augmented matrix we revert to the product:
æ a
11
ç
ç 0
ç
ç 0
ç
çè 0
STROUD
a1,n-2
a1,n-1
a1n
pn-3,n-2
pn-2,n-1
pn-2,n
0
pn-1,n-1
pn-1,n
0
0
pnn
öæ
x
֍ 1
֍ x
֍ 2
֍
֍ x
÷ø è n
ö æ b ö
1
÷ ç
÷
q
÷ ç 2 ÷
÷ =ç
÷
÷ ç
÷
÷ø çè qn ÷ø
Worked examples and exercises are in the text
Programme 5: Matrices
Solution of a set of linear equations
Gaussian elimination method for solving a set of linear equations
æ a
11
ç
ç 0
ç
ç 0
ç
çè 0
a1,n-2
a1,n-1
a1n
pn-3,n-2
pn-2,n-1
pn-2,n
0
pn-1,n-1
pn-1,n
0
0
pnn
öæ
x
֍ 1
֍ x
֍ 2
֍
֍ x
÷ø è n
ö æ b ö
1
÷ ç
÷
÷ ç q2 ÷
÷ =ç
÷
÷ ç
÷
÷ø çè qn ÷ø
From this product the solution is derived by working backwards from the
bottom starting with:
pnn xn  qn so xn 
STROUD
qn
pnn
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Matrices – definitions
Matrix notation
Equal matrices
Addition and subtraction of matrices
Multiplication of matrices
Transpose of a matrix
Special matrices
Determinant of a square matrix
Inverse of a square matrix
Solution of a set of linear equations
Eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Eigenvalues and eigenvectors
Equations of the form:
A.x   x
where A is a square matrix and l is a number (scalar) have non-trivial
solutions (x ¹ 0) for x called eigenvectors or characteristic vectors of A.
The corresponding values of l are called eigenvalues, characteristic values
or latent roots of A.
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Eigenvalues and eigenvectors
Expressed as a set of separate equations:
æ a
11
ç
ç a21
ç :
ç
çè an1
a12
a13
...
a22
a23
...
:
an2
:
an3
...
a1n ö æ x1
֍
a2n ÷ ç x2
: ֍ :
֍
ann ÷ø çè xn
ö
æ x
1
÷
ç
÷ = l ç x2
÷
ç :
÷
ç
÷ø
çè xn
ö
÷
÷
÷
÷
÷ø
That is:
a11x1 + a12 x2 + a13x3 +… + a1n xn = l x1
a21x1 + a22 x2 + a23x3 +… + a2n xn = l x2
:
:
:
:
:
an1x1 + an2 x2 + an3x3 +… + ann xn = l xn
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Eigenvalues and eigenvectors
These can be rewritten as:
æ a -l
11
ç
a21
ç
ç
ç
çè
an1
a12
a13
a22 - l
a23
an2
an3
öæ x
1
֍
a2n ÷ ç x2
֍
֍
ann - l ÷ø çè xn
a1n
ö
æ
ö
÷ ç 0 ÷
÷ ç 0 ÷
÷ =ç
÷
÷ ç
÷ø è 0 ÷ø
That is:
 A I  .x  0
Which means that, for non-trivial solutions:
A  I  0
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Eigenvalues and eigenvectors
Eigenvalues
To find the eigenvalues of:
4
A 
3
solve the characteristic equation:
4 
3
That is:
( 1)(  5)  0
This gives eigenvalues
STROUD
1
2 
1
0
2
1 1; 2  5
Worked examples and exercises are in the text
Programme 5: Matrices
Eigenvalues and eigenvectors
Eigenvectors
4
1
To find the eigenvectors of A  
 solve the equation A.x   x
3
2


For the eigenvalues l = 1 and l = 5
For  =1
 4 1   x1   x1 
 k 
 3 2   x   1 x  and so x2  3x1 giving eigenvector  3k 

 2   2 


For  =5
 x1 
 4 1   x1 
k 

5
and
so
x

x
giving
eigenvector
x 
 3 2  x 
k 
2
1

 2 
 
 2
STROUD
Worked examples and exercises are in the text
Programme 5: Matrices
Learning outcomes
Define a matrix
Understand what is meant by the equality of two matrices
Add and subtract two matrices
Multiply a matrix by a scalar and multiply two matrices together
Obtain the transpose of a matrix
Recognize the special types of matrix
Obtain the determinant, cofactors and adjoint of a square matrix
Obtain the inverse of a non-singular matrix
Use matrices to solve a set of linear equations using inverse matrices
Use the Gaussian elimination method to solve a set of linear equations
Evaluate eigenvalues and eigenvectors
STROUD
Worked examples and exercises are in the text