Transcript A B - Apeiron

MATRIČNA ALGEBRA
Week 2
Matrix - basic definitions
• In this section and the next, we offer
formal definitions
• of matrices and matrix operations, and
state certain important properties of
matrix algebra.
MATRIX -- Basic definitions
You frequently see various types of data
presented in the form of rectangular array
of numbers, such as the box score of a
football, basketball, or baseball game; or
the vital statistics of movies stars,etc.
What is an array?
What is an array?
• An array is rectangular arrangement of
quantities (mathematical elements) in rows and
columns.
• It is another name for a matrix.
MATRIX -- Basic definitions
Consider the following price chart for a
length of pipe of given diameters.
•
2 in. 4 in.
6 in.
•
copper 50¢
65¢
75¢
•
tin
30¢
40¢
45¢
2 in. 4 in. 6 in.
copper 50¢ 65¢ 75¢
tin
30¢ 40¢ 45¢
Such rectangular arrays when subject to
certain operations are example of
matrices..
They are usually designated by capital
letters A , B , C, etc.
Definition. A matrix A is a rectangular
array of entries, denoted by...
 a11 a12 ... a1n 
a

a 22 ... a 2 n 
21

A
 .
.
.
. 


a m1 a m 2 ... a mn 
 a11 a12 ... a1n 
a

a
...
a
21
22
2n 

A
 .
. . . 


a m1 a m 2 ... a mn 
The entries of matrix aij often are called the elements of the matrix.
 a11 a12 ... a1n 
a

a
...
a
2n 
A   21 22
 .
.
.
. 


a m1 a m 2 ... a mn 
Matrix notation allows us to manipulate large
rectangular arrays of numbers as single entities.
This frequently simplifies the statements of various
operations and relationships.
Example 1. Let the matrix
8
AA   1

2
2
1

7

Represent the number of gadgets R, S, and T that factories P
and Q can produce in a day, that is, matrix A represents the
following production capacity.
Factory P Factory Q
Gadget R 8 dnevno 2 dnevno
Gadget S 1 dnevno 1 dnevno
Gadget T 2 dnevno 7 dnevno
Factory P
Gadget R 8 kom/dan
Gadget S 1 kom/dan
Gadget T 2 kom/dan
Factory Q
2 kom/dan
1 kom/dan
7 kom/dan
Later, using the entity “matrix A”, we will extend this
example to accomplish some very useful results.
8
A  
1

2
2
1

7

Scalars?
Scalars...
Scalars!
In this course we shall assume, unless it is
stated otherwise, that the entries of a matrix
are scalars ; (no matrices, no vectors)
We can interpret the term scalar as a complex
number, a real number or a function thereof.
• Under certain conditions, the entries of a matrix
may also be certain other matrices.
Matrix is not a scalar !!!
Matrix does not represent a number of any
type!!
A matrix does not have a numerical value; it is
merely an array.
• Examples of matrices..
The following are examples of matrices:
The entries are integers, functions of x, complex numbers,
real numbers, and matrices.
 2 1 2
4 0 1 


 
 2 
3 5 
2 

 3 
2i i 
 3 i 1


x2 2 


 2 x x  2
 2

4


 6
1

0
1
3
0

1
4 


2 
4 
•Real matrix..
If all entries of a matrix are real numbers, it is called
real matrix.
 2 1 2
4 0 1 


 
 2 
3 5 
2 

 3 
7  5
A 
7 0
2
3 
7
•Zero matrix..
If all of the entries of a matrix are zero, the matrix is called
the zero matrix or null matrix and is denoted by 0. Bold print
will be used to distinguish the zero matrix from zero scalar.
0 0 0 
0

0
0
0


The horizontal lines of the array are called rows.
The vertical lines are columns.
What is a matrix?
• So, we can say,
• A matrix is a rectangular array of numbers (or
scalars) arranged in rows and columns.
Matrix
• Each entry is designated in general as aij , where
i represents the row number, and j is the column
number; thus a31 is the entry in the third row
and first column.
• The double subscript can be called the address
of the entry.
 2 1 2
A 

4 0 1 
The dimensions of the array (number of rows stated first)
determine the order or shape of the matrix, designated
“m by n”.
Matrix A in this example has a total 2 rows and 3
columns of entries. We say that the order (shape) of this
matrix is 2 by 3.
The order of matrix in next example is ?
 
 2 
3 5 
2 

 3 
The order of this
matrix is 3 by 1.
 
 2 
3 5 
2 

 3 
Such a matrix with of a single column is called a column
matrix.
When a matrix consists of a single row (is of order 1 by
n), it is called a row matrix.
6
3
3 314
.

• How many columns does this row matrix have ?
6
3
3 314
.

• Examples..
Example of row-matrix..
•

F  f1 f 2 f 3

Example of matrix
 g11 g12 g13 
G

g
g
g
21
22
23 

What is the address of entry g23 ?
(2,3)
Example of column matrix
 x1 


X   x2 


x
 3
What is the address of entry x2 ?
(2,1)
• Can one subtract or add column and row matrices?
• Why not?
• Because they are not conformable for subtraction and addition.
• Transposition of matrices..
Definition..
• The transpose AT of a m by n matrix A=[aij] is the m by n matrix Q=[qij]
obtained by interchanging the rows and columns of A, thus
• AT=Q=[qij]
• where qij=aji
(i=1,2,…,n; j=1,…,m)
For example..
•
then
if
1  1 3
A
3 7  4
2
5

 1
 1
T
A 
 3

2
3 

7

 4

5 
and..
•
if
 5 
  1

X  
 0 


 4 
then


X  5 1 0 4
T
Several important properties of transposes..
• 1) The transpose of transpose is the original matrix (AT)T=A
• 2) The transpose of a sum is the sum of transposes
(A+B)T=AT+BT
• 3) The transpose of a scalar times a matrix is the same scalar times
the transpose of that matrix (kA)T=kAT
• 4) The transpose of a product is the product of the
transposes in reverse order:
(AB)T = BTAT
• 5) If A and B are two column vectors (ie, row matrix or
column matrix) each of row degree n. Then ATB is a
scalar;hence
• (ATB)T=BT(AT)T=BTA ,
• (ATB)T=BTA
• called inner product, dot product or scalar product.
• Square matrix..
When the dimensions of a matrix are equal, it is called a square matrix.
 3
0

 
4 7 

2 6
0 0 
The main diagonal of a square matrix consists of the entries
a11,a22,a33,…,ann
 3
0



4 7 

2 6
0 0 

Does a non-square matrix have a
main diagonal?
No, it doesn’t!!!
Diagonal matrix..
• A diagonal matrix is a square matrix whose off-diagonal elements, aij
for ij, are equal to zero. Example..
4

A  0
0
0
7
0
0

0
 3
A Unit Matrix, or Identity matrix, or
• 1 Matrix
Identity Matrix
• A unit matrix, or identity matrix, or 1 matrix, is a diagonal matrix
whose diagonal elements are 1. A unit matrix of order n is denoted by
In or simply I.
• For n = 3, we have
1

I 3  0

0
0
1
0
0

0
1

Identity Matrix
• Can a non-square matrix be a unit
matrix?
• No, it can’t, because non-square
matrix has no main diagonal.
• Only square matrices have an identity
matrix!
Identity Matrix
• Hence, the unit matrix is always a square matrix.
This 3 by 3 real matrix is said to be of order 3.
In general an n by n matrix is said to be of order n.
 3
0



4 7 

2 6
0 0 

This is matrix of order 3.
Matrices are denoted in several different ways by
different authors. Some use parentheses, some use
double vertical lines. In this course we will use one of the
three following ways:
 a11 a12
a
a
21
22

A
 .
.

a m1 a m 2
... a1n 

... a 2 n 
 aij
... . 

... a mn 
 
( m,n )
Notation
 9 6 23
A
15 2 0
 12  3 / 4 2.7 


A   655
.  917
. 44


. 012
. 73 
 239
A Matrix does not represent a
number of any type!!!
A Matrix is not a scalar!!!
It does not have a value!!!
Exercises..
Exercises..
• Is the number of rows of a given matrix
always the same as the number of
columns?
• No, it isn’t.
Exercises..
• Is the number of rows of a given matrix always
greater than the number of columns?
• No, it isn’t.
Is this a matrix ?
Why ?
2
1
0
-2
3
No, it isn’t, because is not a rectangular array.
Exercises
• What is the entry in the third row and second column of matrix A?
• What is the address of the entry 6?
 1

A   6

 1
9

2
3

Exercises
• What is a12 and what is a21 in next matrix?
a 
ij
( 2 ,2 )

 2 2

  0 1


 2 3
 

1 4 

9 8 
7 5 


Matrix Equalities and Inequalities
• Definition:
• Two matrices A and B are said to be equal when they are of the same
order (have the same shape) and all their corresponding entries are
equal; that is,
• aij = bij
• for all i and j.
Matrix Equalities and Inequalities
• Example: A=B
 1

A   6

 1
9

2
3

 1

B   6

 1
9

2
3

What is the condition for A=B in this example?
• A=B only when x=3.
x

A  4

1
2

7
3

3

B  4

1
2

7
3

Why does AB in this next example?
x

A  4

1
2

7
3

x

B  4
 1
2
7
3
5

1
9 
AB because A and B do not have
the same order (do not have the
same shape).
Matrix Inequalities
• Definition:
• A real matrix A is said to be “greater than” (>)
real matrix B of the same order when each of
the entries of A is “greater than” each of the
corresponding entries of B.
Matrix Inequalities..
Matrices behave differently from real
numbers with respect to inequalities.
For two real numbers a and b:
If ab and ab,
then a=b.
That is not the case for real matrices
of the same order.
Matrix nonnegativity..
• A matrix is said to be nonnegative if each entry (elements) is said
to be nonnegative if each entry (elements) is nonnegative. Thus
we write
X0
• to indicate that each component xij of X is a nonnegative real
number
• Exercises..
Matrix Equalities and Inequalities
• Find, if possible, all values for each unknown
that will make each of the following true:
1

x

0


4
1
4



y   2 x  1
2 y


3
0
3



Calculate if possible A+B+C=?
1
A
2
4

8
 3  2
C

 2  2 
2 5 
B

0  1
Result:
6 7
A BC  

0 5
Find, if possible, all values for unknown x that will
make of the following true:
2

x

1
 0 1 
3 



 1  6  2 


5 
0
3



• Matrix addition and subtraction..
Matrix Addition
• Matrix addition and subtraction can be performed only when the two
matrices to be added are of the same order.
• We say then that they are
• conformable for addition or
• conformable for subtraction.
Definition:
Given matrices
A=[aij](m,n)
and
B=[bij](m,n)
Matrix addition is defined as
A+B=[aij+ bij] (m,n)
In other words, if two matrices are of the same
order they may be added by adding corresponding
entries.
The same rules are valid for subtracting the matrices.
Matrix subtraction is defined as
A-B=[aij- bij] (m,n)
In other words, if two matrices are of the same order they
may be subtracted by subtracting corresponding entries.
Calculate if possible A+B=?


A 0 5 8
 3 


B    6


4


Not the same order, hence impossible.
Matrix Addition and Subtraction
If A is a 2 by 3 matrix and B is a 3 by 2 matrix, are A and B
conformable for addition?
Are A and B conformable for subtraction?
Why?
Using
1 2 
A

3 4 
2
and B  
1
 1

3
Prove rule 2) for the transpose of the sum of
two matrices :
(A+B)T=AT+BT
Prove rule 2) for the transpose validity for the
case (A-B)T =AT-BT
Example: Matrix Addition
Example:
A manufacturer produces a certain metal.
The costs of purchasing and transporting specific
amounts of necessary raw materials (ores) from
two different locations are given by the following
matrices:
Matrix Addition
p. c. t . c.
16 20 R


A  10 16 S

4
9
 T
p. c. t . c.
12 10 R


B  14 14 S
12 10 T
p.c. means purchasing costs; t.c. means transport costs
R, S and T are ores (rude)
Find the matrix representing the total p.c. and t.c. of each type of ore!
Find the matrix representing the total p.c. and t.c. of each type of ore!
p. c.
28

A  B  24
 21
t . c.
30 R

30 S
14  T
Matrix algebra...
• Week 2-2
• Multiplying a matrix by a scalar...
Definition:
• Given matrix A=[aij](m,n) and scalar k, then
• kA = [k aij](m,n)
• In other words, a matrix may be multiplied by a scalar by multiplying
every entry of the matrix by the scalar.
•
Examples:
3 2 4 
A

1 0  2
and k=2
 2  3 2  2 2  4  6 4 8 
kA



2 1 2  0 2  ( 2) 2 0  4
5 2 6
B

1 0 1
and k=-1
 5
kB  ( 1) B   B  
 1
2
0
 6

 1
• Is there any restriction on the multiplication of a
matrix by scalar?
• No.
• The product of a scalar and matrix is another matrix.
• We are now in position to handle matrix subtraction
similarly to the way in which we perform scalar
subtraction.
• Remember that the rule in real number algebra is:
• a-b = a + (-b) or 6-4=6+(-4)
• Likewise, for matrices, A-B=A+(-B).
Matrix addition is
a) Associative:
(A+B)+C=A+(B+C)
b) Distributive:
k(A+B)=kA+kB
(c+k)A=cA+kA
c) Commutative: A+B=B+A
A+0=A
• Multiplication of Matrices..
The product of two matrices..
• We now define a second kind of product - the product of two
matrices.
• Before giving a general definition, however, we consider the
method of multiplying a row matrix (row-vector) by a column
matrix (column vector).
• (Here the row matrix precedes the column matrix and the
number of columns in A equal number of rows in B)!
Definition
• Let A be a 1 by p matrix and B be a p by 1 matrix. The
product C=AB is a 1 by 1 matrix given by
a
11
a12
... a1 p
b11 
b 
 21 
 . 
 .   a11b11  a12b21 ...a1 pbp1
 
 . 
bp1 



For example,
•

 0
 
2 1 3 4  (2  0)  (1 4)  (3  2)  10 !!!!
2

Product of row matrix and column matrix is 1 by 1
matrix, that is a SCALAR!
Matrix multiplication
• In order to present some logic to the definition of the product of two
matrices, we find it convenient first to define the product of two
vectors.
• The special matrices form consisting of either a single row or a single
column are referred to as vectors.
Matrix multiplication
• A vector, whether it be a row vector or a column vector, is a special
case of the matrix; in the first case it has one row and in the second
case it has one column.
• In either case it can be considered to be a point in n-dimensional
space, where the components are the coordinates.
1 3
The examples of two vectors:
3.5
3
2.5
2
1.5
1
0.5
0
and
 2 2
(1,3)
(2,2)
Series1
0
1
2
3
The examples of two vectors
3.5
3
2.5
2
1.5
1
0.5
0
1
3
 
and
2 
2 
 
(1,3)
(2,2)
Series1

0

1
2
3
Therefore the row vector [a1 a2 a3] or the column
vector
 a1
a
 2

a3





• can be considered to be a point in three-dimensional
space. There is no geometrical distinction between
these two vectors.
• The decision to write a vector as a row or a column
vector is therefore a matter of convenience.
Matrix multiplication
• The operation of addition or subtraction of two vectors or
multiplication by a scalar (as we will se later) follows the rules for
matrices.
• Specific for vectors are inner product and vector length or norm of
vector.
Matrix multiplication
• Notice that the number of columns of A must equal the number of
rows of B.
• (Conformable for multiplication or susceptible to multiplication)
•
One use of this operation may be forecast, if we
observe that a linear equation may be expressed
using a product of two matrices, that is,
x
3     3
y
 
2 
Means [2x+3y]=[3] and therefore
2x + 3y = 3
Now in order to express a system of equations such as
2x+3y = 3,
x + 4y = 1,
using matrix notation, we will need the following
definition.
Here we are not restricted to a row matrix times a
column matrix, although we will see that the
multiplication simply requires a succession of the
manipulation previously described.
Definition..
• Let A be an m by p matrix and B be a p by n matrix.
• The product C=AB is an m by n matrix where each entry cij of C is
obtained by multiplying corresponding entries of the ith row of A by
those of the jth column of B and the adding the results.
Matrix multiplication
• The operation defined above can be illustrated in general by the
following diagrams.
 a11

 a21


am1
a12
a22
. .
am 2
... a1 p  b11 b12

... a2 p  b21 b22
 . .
. .

... amp  bp1 bp 2
... b1n   c11 c12
 
b
... 2 n  c21 c22

. . . .
 
... bpn  cm1 cm2
... c1n 

... c2 n

. .

... cmn 
Matrix multiplication
• Where
• c11=a11b11+a12b21+…+a1pbp1
• or, generally speaking...
Matrix multiplication
a
11

 .
 ai1
 .

am1
...
.
...
.
...
a1 p   b11

. 
aip  


.

amp  bp1
... bij
. . .
. . .
. . .
... bpj
... b1n   c11 ... c1n 



. .
.
.
.
 

. .    . cij . 
. .   . . . 
 

... bpn  cm1 ... cmn 
Matrix multiplication
• Where
• cij = ai1b1j + ai2b2j +…+aipbpj
One very handy rule..
• If you multiply matrix A of dimension (m,n) and matrix B of dimension
(n,k), then result matrix will be matrix C of dimension (m,k).
• You can see that (m,k) are the outer numbers of matrices A and B
dimensions.
Example:
•
1 2 
A

1 0
3 0 1
B

 0 1 1
1 2  3 0 1
AB  



1 0  0 1 1
(1 3  2  0) (1 0  2  1) (1 1  2  1)  3 2 3
AB  



(1 3  0  0) (1 0  0  1) (1 1  0  1)  3 0 1
Find result BA, if possible ?!
Why is this impossible?
• This is because the number of columns of the left
matrix does not equal the number of rows of the
right matrix.
• In order for multiplication of matrices to be
performed, the number of columns of the left matrix
must equal the number of rows of the right matrix.
We than say that the left matrix is conformable or
susceptible for multiplication to the right matrix.
Matrix multiplication
• Prior to our definition of matrix multiplication we implied that it
could be used to express the system of equations
• 2x + 3y =3
• x + 4y =1 Consider the following matrix
•
equation.
•
2
1

3  x 
3
 



4  y 
1
2x + 3y =3
x + 4y =1
2
1

3  x 
3
 



4  y 
1
• Perform the indicated matrix multiplication on the left side;there
results
2 x  3 y 
3
 x  4 y   1


 
By the definition of the equality of matrices we
obtain our original system.
Example
2
AB  
3
1 0


2 1
0 4 
2 1 
B
A


1
3
3
2




4  (2  0  11) (2  4  1 3)   1 11





3  (3  0  2 1) (3  4  2  3) 2 18
To form A, B was premultiplied by A;To form BA,
B was postmultiplied by A.
0
BA  
1
4  2


3  3
1 (0  2  4  3) (0 1  4  2) 12




2  (1 2  3  3) (11  3  2)  11
Note: BAAB. This is quite different from scalar
algebra where ab=ba (that is 23=32).
8

7
Matrix Multiplication
When we say matrix B is postmultiplied by matrix A, we mean BA.
When we say matrix B is premultiplied by matrix A, we mean AB.
Using:
1 2 
A

3 4 
2
B
1
 1

3
Proof that the transpose of a product two matrices is the product of
the transposes in reverse order,e.g.
(AB)T=BTAT
Positive integral powers of square matrices..
we define them as we did for scalars
A2 = AA
A3 = AAA
etc.
• Could you realize why A has to be square for An (for
n2) ?
Only then does A have the same number of
columns as the number of rows, necessary for
matrix multiplication.
One characteristic of the identity matrix..
If I is the unit matrix, and A is a square matrix of the same order, then
IA=AI
Could you realize why A has to be square matrix in this case?
One definition..
A diagonal matrix is a square matrix whose off-diagonal elements, aij
for ij, are all equal to zero.
Example:
An identity matrix is a diagonal matrix.
Example...
One illustration of how matrix multiplication my be used.
A simple problem from the field of decision making.
Example..
A certain fruit grower in Florida has a boxcar loaded with fruit ready to
the shipped north. The load consists of 900 boxes of oranges, 700
boxes of grapefruit, and 400 boxes of tangerines. The market prices,
per box, of the different types of fruit in various cities are given by the
following chart.
Example..
oranges grapefruit tangerines
New York
$4 per box
$2 per box $3 per box
Cleveland
$5 per box
$1 per box $2 per box
St. Louis
$4 per box
$3 per box $2 per box
Oklahoma City $3 per box
$2 per box $5 per box
To which city should the carload of fruit be sent in
order for the grower to get maximum gross
receipts for this fruit?
Solution...
Consider the chart..
oranges
New York
$4 per box
Cleveland
$5 per box
St. Louis
$4 per box
Oklahoma City $3 per box
grapefruit
$2 per box
$1 per box
$3 per box
$2 per box
tangerines
$3 per box
$2 per box
$2 per box
$5 per box
as the “price matrix”, and form the quantity matrix
900boxes


700boxes

400boxes

Example..
The product of these matrices, as shown below, yields an “income
matrix” where each entry represents the total income from all the fruit
at the respective cities.
4
5

4

3
2 3
3600  1400  1200 6200
900 

1 2   4500  700  800  6000
 700  


3 2    3600  2100  800  6500
 400 
 

2 5
2700  1400  2000 6100
Example..
4
5

4

3
2 3
3600  1400  1200 6200
900 




1 2   4500  700  800
6000
 700  
 
3 2    3600  2100  800  6500
 400 
  
2 5
2700  1400  2000 6100
The largest entry in the income matrix is 6500,
and therefore the greatest income will come
from St. Louis.
Exercises..
Given V1 and V2. Find V1 V2.
V1  3 0 6 2
V2
  1
 2 

 
 3 


 7 
V1V2=3(-1)+02+6 3+2 7=29
Multiply...
2
A 
4

0
3 
and
 1

2 

3 0 6 4 
B

1 2 5  3
•
 9 6 27  1 


A  B  11  2 19 20
 2 4 10  6 
Matrix multiplication laws..
The associative law for multiplication is
• ABC=(AB)C
which means that if we wish to find the product of three matrices, we
can either multiply the first two matrices and then postmultiply it by
the third matrix, or we can multiply the second and third matrices and
then premultiply it by the first matrix.
Matrix Multiplication
The distributive law for multiplication with respect to addition requires
that identical results be obtained whether you add the products AB and
AC or multiply A by the sum of B and C:
A(B+C)=AB+BC
...
• -Sigma notation
• -Elementary matrix transformation
• -Determinants
• -Inverse matrix
• -Gauss-Jordan operations
• -Solution of simultaneous system of equations
• - Cramer’s rule
Sigma notation
• When matrix multiplication is discussed in general, the co called “
notation” (read sigma notation) is very helpful.  is letter from Greek
alphabet and in mathematics usually stands for “sum of”.
• For example, the sum
ores
5
1 2 3 4 5  k
2
2
2
2
2
2
k 1
The expression is read “the sum of k2 where k ranges
from 1 through 5”. k is called the index of summation.
Examples:
2+4+6+8+10+…+98+100=
50
 2k
k 1
100
x1+x2+x3+…+x100=
x
k 1
k
3
ai1+ai2+ai3=
a
k 1
ik
3
a11b11+a12b21+a13b31=
a
k 1
b
1k k 1
3
a11b11+a12b21+a13b31=
a
k 1
b
1k k 1
• This example my be recognized as the entry in the first row and first
column of the product of the two matrices.
 a11

 a21


am1
a12
a22
. .
am 2
... a1 p  b11 b12

... a2 p  b21 b22
 . .
. .

... amp  bp1 bp 2
... b1n   c11 c12
 
b
... 2 n  c21 c22

. . . .
 
... bpn  cm1 cm2
... c1n 

... c2 n

. .

... cmn 
Elementary matrix transformation..
Definition: The three operations:
1) interchange any two rows, (this is equivalent to
writing the equation in a different order and obviously
this does not effect the solutions of the system),
2) multiply any row by a nonzero scalar,
3) add to any row a scalar multiple of another row,
are called elementary row operations.
Elementary Operations
• Elementary column operations are defined by replacing the
word “row” by “column” throughout the preceding
definition.
• An elementary operation is any operation that is either an
elementary row operation or an elementary column
operation.
Equivalent matrices
If a matrix A can be transformed into a matrix B by means of one or
more elementary operation, we write
AB
and say that A is equivalent to B.
Example.. The matrix
 1
 2

3
3
4
1

• Can be transformed to
1 3 4 
0 9 9 
• by elementary row


• operation 3) - multiply the first row by 2 and add to
second row (R2=2R1+R2). This then can be transformed to
3 4 the second row by
12) -multiply
• by elementary operation
0 1 1 
1/9 (R2=R2/9).


•
2
4

Exercises..
1
5


,
2
0

1
3

R2=(-2R1+R2)
• Which elementary row operation (if any)
transforms the first matrix into the second?
2
4

1
5

-2R1
2
4

1
5

+R2
-4
-2
Which elementary row operation (if any)
transforms the first matrix into the second?
•
2
1

4
2
6
1


1
4

2
2
3
4

Determinants..
• A matrix has no numerical value.
• However, every square matrix of scalars has what is called
determinant.
Determinants..
• The Determinant is a number associated
with every square matrix.
• The determinant of A, denoted by
• |A|, is obtained as the sum of all possible
products in each of which there appears
one and only one element from each row
and each column of A.
The determinant of A is..
a11 a12 ... a1n
A
am1
. . ... .
. . ... .
am2 ... amn
Finding the determinant of a 2 by 2 matrix is
straightforward..
a11 a12 
A

a21 a22 
•
+
-
a11 a12
A
 a11a22  a21a12
a21 a22
Determinants..
• For a larger-order determinants, define Dij
• as the determinant of the [(n-1)x(n-1)] matrix formed by striking out
(or deleting) the ith row and jth column of A.
• Dij is called the (i,j) “minor” of A.
For example: minor D11 of |A|...
•
a11 a12 a13
A  a21 a22 a23
a31 a32 a33
a22 a23
D11 
a32 a33
Next, we define the (i,j) cofactor of A to be
• ij=(-1)i+j Dij
• that is, the (i,j) minor ( Dij ) with a negative sign attached if the sum of
subscripts is odd, and a positive sign if the sum is even.
• Examples: 11= (-1)2D11, 12=(-1)3D12, 13=14D13, etc.
Determinants..
• The minor Dij of the element aij is the determinant obtained from the
square matrix A by striking out (or deleting) the ith row and jth
column.
So,
a11
a12
a13
A  a21
a22
a23
a31
a32
a33
+
+
a11 a12
A  a 21 a 22
a31 a32
a13
a 22
a 23  a11
a32
a33
a 23
a 21
 a12
a33
a31
a 23
a 21
 a13
a33
a31
a 22
a32
Example..
11 4 6
A11  (1)
3 2
 3
A
 2

 1
5
4
3
2 6
A12  (1)
1 2
1 2
7
6

2

A13  (1)
1 3
2 4
1 3
4 6 2 6 2 4
A 3
5
7
 3(10)  5(10)  7(10)  10
3 2 1 2 1 3
Properties of determinants
• 1) If a matrix B is formed from a matrix A by the
interchange of two parallel lines (rows or columns)
then |A|=-|B|.
• 2) The determinants of a matrix and its transpose are
equal; that is, |A|=|AT|.
• 3) If all of the entries of any row or column of matrix
A are zero, then |A|=0.
• 4) The determinant of a matrix with two identical
parallel rows or columns is zero.
Example for properties of 3) and 4)
•
0 1 4
2 2 1
A  2 1 6 0
A  3 3 1 0
0 0 0
4 4 0
•Inverse matrices...
Inverse matrix..
• The inverse of a matrix A of order (n x n) is the matrix A-1
such that AA-1=I, or A-1A=I.
• In other words, the matrix which when multiplied by the
matrix A produces the identity matrix is said to be inverse of
A and is denoted by A-1.
• Since A conforms to A-1 and A-1 conforms to A, A must be
square. Therefore, we conclude that only square matrices
(but not all, f.e. singular) have inverses.
The singular matrix
• The singular matrix is matrix which determinant is equal zero.
Inverse Matrices
• Division is not defined in matrix algebra.
• The matrix algebra counterpart of division in scalar
algebra is achieved through matrix inversion.
• In algebra of real numbers, the inverse of of X is 1/X,
is a quantity which when multiplied by X yields 1.
Inverse Matrix
• The inverse of a square matrix A, which we designate A-1, is a matrix
which when postmultiplied or premultiplied by A yields the unit or
identity matrix.
• AA-1=I, A-1A=I.
• Or, we can say, if two square matrices A and B, each of degree p,
satisfy the equation AB=Ip we call B the inverse of A, written B=A-1.
Proof..
• This is valid AA-1=I, A-1A=I.
• Suppose is given a matrix B which satisfies BA=I.
• If we now multiply each side of the last equation by A-1 on
the right, we have
• BAA-1=IA-1
• which reduces to B=A-1.
Matrix Inverses
• Thus, the inverse of matrix A is..
1
 2 3 1   1 / 3  1 / 9 5 / 9 




1
A  0 1 2    2 / 3  1 / 9  4 / 9 
3 2 1   1 / 3 5 / 9 2 / 9 
How we calculate A-1?
 a11 a12 
A

a21 a22 
• The inverse of any square matrix A consisting of two
rows and two columns is computed as follows:
a
/
d

a
/
d


22
12
1
A 

  a21 / d a11 / d 
Where d=a11a22-a21a12 is determinant of the 2x2 matrix
A.
Example..
0.7  0.3
A

 0.3 0.9
Then d=0.70.9-(-0.3)(0.3)=0.54 and
0
.
9
/
054
.
0
.
3
/
054
.
167
.
056
.




1
A 



. 0.7 / 054
.  056
. 130
. 
0.3 / 054
Gauss-Jordan operations..
• One way to form the inverse of a square matrix is to
perform simultaneously on a unit matrix, of the same
order, necessary row operations to convert the left
matrix to I. This is known as Gauss-Jordan operations.
Thus, given
2 4 
A

3 2 
Gauss-Jordan operations..
• Arrange alongside A a unit matrix I:
2 4 
1 0
 3 2   0 1




2 4 
1 0
 3 2   0 1




• Now perform on A the row-column operations via which
the Gauss-Jordan procedure is employed to convert A to
a unit matrix-performing these operations
simultaneously on I.
• Thus, when the first row is divided by 2 to yield a 1 in
the first row, first column of A, divide the first row of I
by 2 also (R1=R1/2). When the second row of A is replaced
by the sum of itself and the new first row multiplied by 3, replace the second row of I similarly (R2=R2+(-3)R1).
January 21, 1999
Top Ten Ways to Succeed in MSCI 2400
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•
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•
•
•
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Understand –Don’t Memorize
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Gauss-Jordan operations..
• The final result is
1
0

IA
1
0
 1 / 4 1 / 2



1
3 / 8  1 / 4 
Thus, as A is transformed into I, I is transformed
into A-1.
Why is the inverse matrix so important?
• To get some appreciation and understanding of the
utility of the matrix inverse, consider the following
linear system of equations. Let
15
.
1 2.4 


A 1 5 1 

.
3 3.5
15

 x1 


X   x2 

 x3 

Thus if we form AX=b
2000


b  8000
5000
15
.
1 2.4 


A 1 5 1 

.
3 3.5
15

 x1 


X   x2 

 x3 

2000


b  8000
5000
• we really have the following system of three
equations in three unknowns:
• 1.5x1+ x2+ 2.4x3=2000
• x1+5x2+ x3=8000
• 1.5x1+3x2+3.5x3=5000
• To obtain a solution to this system when written in
the matrix form, we start with the equation AX=b
AX=b
• Multiplying both sides of the equation on the left by
A-1, we obtain
• A-1AX=A-1b
• Evaluating the left-hand side by using the associative
law, we have
• A-1AX=(A-1A)X=IX=X
• and thus our solution is
• X=A-1b
• According to X=A-1b, then, to solve m equations in m
unknowns, we should find the inverse of the matrix
of coefficients and multiply this by the column vector
of constants b.This result will be a column- vector X.
• Thus, if we are able to find the inverse of the matrix
of coefficients, it is easy to find the solution of the
system of equations.
15
 x1 
.
1 2.4 


 
A   1 5 1  X  x2
 

.
3 3.5

15

 x3 

2000


b  8000
5000
• AX=b, according to X=A-1b, we should find the inverse
of the matrix A and multiply this by the column
vector b. The result will be a column vector X.
15
.
1 2.4  1


 1 5 1  0

.
3 3.5
15

0
0
1
0
0

0
1

• Now we should to perform next elementary rowcolumn operations to get A-1:
15
. 1 2.4 1 0 0



 1 5 1  0 1 0
15
. 3 35
.  0 0 1
• R1=R1/1.5
• R2=R2-R1
• R3=R3-1.5R1
• R2= R2/4.333
• R3= R3/1.37692
• R2=0.13846R3+ R2
• R1=-1.69231R3+ R1
  16201

.
0.4134  12291
.

1 
A    0.2235 01844
.
01006
.

 05028
.
 0.3352 0.7263
• After that, using X=A-1b we will calculate
• solution vector X.
  16201
.
0.4134  12291
.  2000  4019
. 

  

1
X  A b    0.2235 01844
.
01006
.
.
  8000  15319
 05028
.
 0.3352 0.7263 5000   557
. 
• The best computational scheme for calculating of a
solution of n equations with n variables (vector Xp) is
to follow a procedure similar to this described earlier
for finding the inverse matrix. But, instead of
computing the inverse matrix, we may set up
together A and b and then apply the necessary row
transformations to reduce A to its identity form.
•
A
b
I
A-1b
• In other words, by applying the same transformations to b that we
did to A, we have converted b into A-1b which is solution vector.
• In this manner we compute A-1b without actually finding A-1 and
without performing the multiplication of A-1 by b.
• If for any reason we should want the inverse of A, we
can get it by setting up I behind A on this manner:
I operations,
A proper row
b
• After the
we will have
I
A-1
A-1b
Inverse matrix..
• Note that all matrices do not have inverses, yet
whenever the inverse to a matrix can be found, it is
unique.
But, the existence of a solution is not dependent on
the existence of the inverse matrix. Reasons for that
almost are: redundancy, inconsistency or singularity
of the set of equations.
Redundancy
• Two or more equations of the set may be linearly
dependent (that is, a combination of sums or
differences of multiples of other equations).
• The dependent equations are redundant and can be
omitted from the system. We then have a system of
fewer than m equations in m variables. (Many
solutions).
Example of one redundant system..
• 2x1 -4x2 + x3=7
• -x1+7x2 -3x3=12
• 4x1 -8x2+2x3=14
Inconsistency
• On the other hand, system can be inconsistent and
then A-1 does not exist.
• For example, consider:
• 3x1+2x2+ x3=14
• 2x1 -x2+3x3=10
• 5x1+ x2 +4x3=5
• Here the third equation is inconsistent with the first
two and no solution exist.
Singularity
• Finally, a matrix could be singular and its inverse
does not exist.
• 3x1+3x2+ x3=14
• 2x1+2x2+2x3=10
• 5x1+5x2 -2x3 =7
• x1 and x2 have equal coefficients in each equation
and, hence, are dependent on each other. The matrix
of coefficients is then singular and its inverse does
not exist.
Note!
• Every nonsquare matrix is singular.
Exercise..
• Compute the inverse matrix of A:
 4 8
A

6 4 
R1/4; (-6R1+R2); R2/8; (-2R2+R1)
 4 8
A

6 4 
• First set the problem in the following form:
4
6

1
6

A
I
8
4 
1 0
0 1


2
4 
1 / 4 0
 0 1 


 1 2   1/ 4 0 
0  8  3 / 2 1

 

Now perform the G-J steps
on both of these matrices
simultaneously. (First R1/4)
Now multiply R1 by -6 and
add result to the second
row. (-6R1+R2)
Next divide R2 by -8. (R2/-8)
 1 2   1/ 4 0 
0  8  3 / 2 1

 

•
1 2   1 / 4 0 
0 1 3 / 16  1 / 8

 

I
A-1
1 0   1 / 8 1 / 4 
0 1 3 / 16  1 / 8

 

Finally, (-2R2+R1)
Solution of a set of simultaneous linear equations
• Any set of simultaneous linear equations has a
convenient representation using matrix notation. The
system
• a11x1+a12x2+…+a1nxn = b1
• a21x1+a22x2+…+a2nxn = b2
•… … … … … …
• am1x1+am2x2+…+amnxn = bm
• can be written as AX=b
where..
• A=[aij](m,n)
 x1 
 . 


X   . 
 . 


x
 n
 b1 
 . 


b   . 
 . 


b
 m
If A is square (m=n) and nonsingular,
the solution vector is given:
• 1) by X=A-1b (using matrix inverse notation) or
• 2) using Cramer’s rule for nonsingular determinants
i
det( A)
xj 
, j  1,2,..., n
det A
• where (iA) denote the matrix obtained from A by
replacing the jth column of A by the vector b.
Example of Cramer’s rule application..
• 2x1+x2+ x3=0
• x1 -x2+5x3=0
•
x2 - x3=4
x2 
x1 
0
1
1
0
1
5
4
1
1
2
1
1
1
1
5
0
1
1
2
0
1
2
1
0
1
0
5
1
1
0
0
4
1
0
1
4
2
1
1
2
1
1
1
1
5
1
1
5
0
1
1
0
1
1
x3 
•
Homework 1
January 19,1999
• 1) Express the product of 2 by 2 matrices using notation.
2) Write the following without  notation:
5
k
k 1
7
 ( k  2)
k 3
4
a
k 1
k
3
a
k 1
2k
ak 3
3) Express the sums in sigma notation:
a11 a12  b11 b12

a

 21 a22  b21 b22
b13 

b23 
4) Find by definition determinant of the matrix:
2

A  4
0
1
8
7
3

6
5
5) Using matrix A given below proof that the
determinants of a matrix and its transpose are equal.
1

0

0
2
2
0
3

1
4

6) Using elementary matrix operation suggested
below (or self selected), find the solution (vector X)
of the next system:
15
.

1
15
.
1
5
3
2.4   x1 2000
   

1    x 2   8000
3.5   x 3 5000
R1=R1/1.5
R2=R2-R1
R3=R3-1.5R1
R2= R2/4.333
R3= R3/1.37692
R2=0.13846R3+ R2
R1=-1.69231R3+ R1
• 7) Using Cramer’s rule find solution of next system of simultaneous
equations:
• 2x1+x2+ x3=0
• x1 -x2+5x3=0
•
x2 - x3=4