Chapter 2 Solving Linear Systems
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Transcript 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:
aa1 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