econ 1150, spring 2013 linear algebra matrix operations special matrices lecture 6

ECON 1150, Spring 2013

Linear Algebra

Matrix Operations

Special Matrices

Lecture 6

1. Systems of Linear Equations

Consider the following equations: x1 + 2x2 = 2 (1) 2x1 – x2 = 4 (2)

Method of substitution

Method of elimination

Solution:

(x1*, x2*)


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?


x1* = 2, x2* = 3, x3* = 4

Row operations:

•Multiply an equation by a nonzero constant

•Add a multiple of one equation to another

•Interchange any two equations

Gaussian Elimination


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.


The following systems are linearly dependent.

2x + y = 8 (1)

x – y = 1 (2)

-1.5x + 3y = 1.5 (3)

– x – y + z = – 2 (4)

3x + 2y – 2z = 7 (5)

x – y + z = 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, x – y = 4, 2x – 3y = 6ECON 1150, Spring 2013

General equation system

Matrix algebra can help to

a. simplify the expression

b. solve the system efficiently

c. testing the existence of a solution

nnn2n21n1

33n232131

22n222121

11n212111

xxx

xxx

xxx

xxx

baaa

baaa

baaa

baaa

n

n

n

n


2. Matrices and Vectors

A matrix is a rectangular array of numbers.

=


Order (dimension) of a matrix

= (# of rows) x (# of columns)

Row vector: a matrix with only one row

Column vector: a matrix with only one column[3 5 -6 1]

752

143 ,

43

21YX

2x2 2x3


Two matrices are equal if they have the same order and the corresponding elements are equal. E.g.,

x = y = 2.

20

1

2

21 y

yx2x2 2x2


3. Matrix Operations

Addition and subtraction

Adding up elements of the same corresponding position

Conformability: Same order

Example 6.1:

NMNMNM

CBA

?356

143

85

20 3.

?04

52

13

21 2.

?04

52

13

21 1.


Scalar multiplication

Conformability: Not required

32.52

1.510.50.5

12108

6422

654

321

A

A

A


Multiplication

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.

PMPNNM

CBA


PMPNNM

CBA

32635241

6

5

4

321

1x3

3x1

1x1

5032938271 635241

96

85

74

321

1x3

3x2

1x2


PMPNNM

CBA

122

32

695847

635241

6

5

4

987

321

2x33x1

2x1


1222

2

7B ,

617

114A

Let C = AB = . Then

2

1

c

c

c1 = 4(7) + 11(2)

2

7114B ,

617A

c2 = 17(7) + 6(2)

2

7

617B ,

114A

131

50

26717

21174

2

7

617

114AB


20

11

34

B ,112

015A

22

2332

59

1621

213

112112

213

2

1

3

112

014

015001145

0

1

4015

BA


Remark:

• (AB)C = A(BC)

• A(B + C) = AB + AC

• (A + B)C = AC + BC


4. Special Matrices

4.1 Square Matrices

# of rows = # of columns

E.g.,

465

184

327

B ,3324

1222A


4.2 Identity Matrices

100

010

001

10

012 3I I ,

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

4.5 Symmetric matrices

A square matrix that is equal to its transpose.

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.

49

08

13TA

401

9-83A ,

TA

3

2-

1

A

65-

64

5-4


econ 1150, spring 2013 linear algebra matrix operations special matrices lecture 6

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