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Lecture 6
Linear Algebra
Matrix Operations
Special Matrices
ECON 1150, Spring 2013
1. Systems of Linear Equations
Consider the following equations:
x1 + 2x2 = 2
(1)
2x1 – x2 = 4
(2)
Method of substitution
Solution:
Method of elimination
(x1*, x2*)
ECON 1150, Spring 2013
Consider the following equations:
4x1 – x2 + 2x3 = 13
(1)
x1 + 2x2 – 2x3 = 0
(2)
-x1 + x2 + x3 = 5
(3)
How is the solution (x1*, x2*, x3*) found?
ECON 1150, Spring 2013
Row operations:
•Multiply an equation by a nonzero constant
•Add a multiple of one equation to another
•Interchange any two equations
Gaussian Elimination
x1* = 2, x2* = 3, x3* = 4
ECON 1150, Spring 2013
General rule: To solve a system of linear
equations with a unique solution, the
number of linearly independent equations
must be equal to the number of variables.
If any equation of a system of equations can
be derived from a series of row operations
on that system, then this system is called
linearly dependent. Otherwise, it is called
linearly independent.
ECON 1150, Spring 2013
The following systems are linearly
dependent.
2x + y = 8
(1)
x– y=1
(2)
-1.5x + 3y = 1.5
–x –
y +
z =–2
3x + 2y – 2z =
x –
y +
(3)
z =
(4)
7
(5)
4
(6)
ECON 1150, Spring 2013
Equation system:
m linearly independent equations
n unknowns
Case 1: m = n  unique solution
x + y = 10, x – y = 6
Case 2: m < n  Infinitely many solutions
x + y = 10
Case 3: m > n  No solution.
x + y = 10,
–Spring
y =20134, 2x – 3y = 6
ECON x
1150,
General equation system
a11x1  a12 x 2    a1n x  b1
a21x1  a22 x 2    a2n x  b 2
a31x1  a32 x 2    a3n x  b 3

an1x1  an2 x 2    ann x  b n
Matrix algebra can help to
a. simplify the expression
b. solve the system efficiently
ECON 1150, Spring 2013
c. testing the existence
of a solution
2. Matrices and Vectors
Cars required
Week 1 Week 2 Week3
4
7 2
3
5 5
Compact
4
7
2
Intermediate
3
5
5
12
9
5
People carrier
2
1
3
2
1 3
Luxury limousine
1
1
2
1
1 2
Large
A =
12 9 5
A matrix is a rectangular array of numbers.
ECON 1150, Spring 2013
Order (dimension) of a matrix
= (# of rows) x (# of columns)
1 2
3 4 1 
X 
, Y 


3 4
2 5 7 
2x2
Row vector: a
matrix with
only one row
[3 5 -6 1]
2x3
Column
vector: a
matrix with
only one
ECON 1150,
Spring 2013
column
9
8
2
3
Two matrices are equal if they have the
same order and the corresponding
elements are equal. E.g.,
 1
x  y

2 1


2  0
2x2
y

2
2x2
x = y = 2.
ECON 1150, Spring 2013
3. Matrix Operations
Addition and subtraction
A
M N

Adding up elements of the same
corresponding position
Conformability: Same order
Example 6.1:
1 2 2  5
1. 

?


3 1 4 0 
1 2 2  5
2. 

?


3 1 4 0 
 0 2 3 4 1 
3. ECON
 5 1150,
 Spring
6 52013
?
8

3

 

B
M N

C
M N
Scalar multiplication
Conformability: Not required
1 2 3 
A

4
5
6


2 4 6 
2A  

8 10 12 
0.5 1 1.5 
0.5 A  

 2 2.5 3 
ECON 1150, Spring 2013
A
M N
Multiplication
B
N P

C
M P
Conformability: The number of the columns
of A is equal to the number of rows of B.
Rule of matrix multiplication:
The element of the ith row and jth column of
matrix C
ith row of A

jth column of B
and adding the resulting products.
ECON 1150, Spring 2013
A
M N
B
N P

C
M P
 4
1 2 3 5  1 4  2  5  3  6  32
1x1
1x3 6
 
3x1
4 7
1 2 3 5 8 
1x3 6 9
3x2
1 4  2  5  3  6
1 7  2  8  3  9  32
50 
1x2
ECON 1150, Spring 2013
A
M N
B
N P

C
M P
 4
1 2 3   1 4  2  5  3  6   32 
7 8 9 5  7  4  8  5  9  6  122 

 6 
  
 
2x3
2x1
3x1
ECON 1150, Spring 2013
 4 11
7 
A
, B 

17 6 
 2
22
Let C = AB =
 c1 
c  .
 2
c1 = 4(7) + 11(2)
c2 = 17(7) + 6(2)
21
Then
 4 11
7
A
, B 

17 6 
2
 4 11
7 
A
, B 

17 6 
2
 4 11 7  47   112  50 
AB  

 




7  62 131
17 6  2ECON
 1150,
17
Spring 2013
 4 3
5 1 0 
1 1
A
,
B




2
1

1


0 2
4 3 
5 1 0  

A B 
1
1



23 32
 2 1  1 0 2 


5 4   11  00  5 3   11  02 














2
4

1
1

1
0
2
3

1
1

1
2


21 16


9
5


22
ECON 1150, Spring 2013
Remark:
• (AB)C = A(BC)
• A(B + C) = AB + AC
• (A + B)C = AC + BC
ECON 1150, Spring 2013
4. Special Matrices
4.1 Square Matrices
# of rows = # of columns
E.g.,
2 1
A 
,

22
4 2
7 2 3


B  4 8 1 
33
5 6 4
ECON 1150, Spring 2013
4.2 Identity Matrices
 1 0 0
I3  0 1 0
0 0 1
 1 0
I2  
,

0 1
For any M by N matrix A,
IMA = AIN = A.
4.3 Null Matrices
A matrix whose elements are all zero.
ECON 1150, Spring 2013
4.4 The transpose of a matrix A is a new
matrix AT such that the ith row of A is the ith
column of AT.
3 8 - 9
A
,

1 0 4 
 3 1
A T   8 0
 9 4
4.5 Symmetric matrices
A square matrix that is equal to its transpose.
 1 4 - 5
A   4 - 2 6   A T
- 5 6 ECON31150,
 Spring 2013