Download Chapter 2 Solving Linear Systems

Chapter 2 Solving Linear Systems
• Matrix Definitions
– Matrix--- Rectangular array/ block of numbers.
 1 0  20 


 0 1  15 
 1 1 0 


0 


 500 
0 


– The size/order/dimension of a matrix:
• (The numbers of ROWS) by(x) (the numbers of COLUMNS)
– ELEMENTS: individual numbers of matrix
– aij --- an element of ROW i and COLUMN j
– SQURE matrix
• The numbers of ROWS = the numbers of COLUMNS
– IDENTITY matrix: symbol---I
– TRANSPOSED matrix: Rows and columns of
a matrix are switched
–
 1 4
1 2 3 

A  
 4 5 6


A   2 5
 3 6


t
• Matrix Operations
– Addition
• Two same size matrices can be added.
• C=A+B=B+A
 1 2 3


A   4 5 6
7 8 9


10 11 12 


B  13 14 15 
16 17 18 


 1  10 2  11 3  12   11 13 15 

 

C   4  13 5  14 6  15    17 19 21 
 7  16 8  17 9  18   23 25 27 

 

– Multiplication
• Multiplication of a Matrix by a Scalar
– A=kA
– Example
• Multiplication of 2 Matrices
– Two Matrix can be multiplied if and only if--The NUMBER OF COLUMNS OF THE FIRST MATRIX = The
NUMBER OF ROWS OF THE SECOND MATRIX
– The Size of the resultant matrix --the NUMBER OF ROWS OF THE FIRST MATRIX by the
NUMBER OF COLUMNS OF THE SECOND MATRIX
• Example
First Matrix Second Matrix
A
(a）（2x2)
(b）（3x3)
(c）（3x3)
(d）（5x5)
B
(2x2)
(3x2)
(2x3)
(5x1)
Multipication
Possible?
AB
YES
YES
NO
YES
Size
(2x2)
(3x2)
(5x1)
• Notice that:
– AB exists and so does BA with BA being (2x2)
– AB exists, BA does not exist as a (3x2) cannot be
multiplied into a (3x3)
– AB does not exist, It’s possible that BA exists
• How to calculate the elements of C=AB
– Example
 1 2 3


A   4 5 6
7 8 9


 68 


C  AB   67 
 266 


10 
 
B   11 
12 
 
1
A  
3
2
5

B  
4
7
 19 22 

C  AB  
 43 50 
 23 34 

C  BA  
 31 46 
AB  BA
6

8
– A---mxn matrix
» IA=A
» AI =A
I=identity matrix
– Matrix Inversion
• Only Square matrices have the inverse but not all
square matrices have inverses.
• Scalar number:
aa 1  1  a 1a
•
1
a 
a
The inverse of matrix A is denoted by A-1
The size of A-1 is the same as A and
A A-1 = I = A-1 A
Any Matrix times its own inverse is just the
appropriately sized identity matrix
1
•
•
•
•
– Matrix Equality
• Two matrices are said to be equal if
– They are same size
– Corresponding elements in the two matrices are the same
• Break-Even Model in Matrix Algebra terms
– Break-even model in linear equations
1 TR + 0 TC – 20q = 0
0 TR + 1 TC – 25q = 500
1 TR – 1 TC + 0q = 0
– Let
 1 0  20 
 TR 
0 


 


A   0 1  15  x   TC  b   500 
 1 1 0 
q 
0 

3x3
 3 x1 
3 x1
A3x3 x3 x1  b3 x1



– Example
– Ax=b
A-1 Ax= A-1 b
I x= A-1 b
x= A-1 b
4
4  0   2000 
 -3


 

1
x  A b   -3
4
3  500    2000 
 - 0.2 0.2 0.2  0   100 


 

– Modelling Steps
• Set up the system of linear equations
• Decide upon an order in which to express the
unknowns
• The unknowns on the LHS of the equations
• Identify the following 3 matrices
– A: Square matrix of coefficients relating to the unknowns
– x: the matrix of unknows
– b: the matrix of RHS constants
•
•
•
•
Find matrix inverse A-1 of A
Perform the matrix multiplication A-1b
Use the matrix equality rule to find the elements of x
Give the business interpretation of x