LESSON 2 MATRICES

1
LESSON 2 MATRICES
2
Basics of Matrices

• A matrix is a rectangular array of ordered
  numbers.
• Example. Let A denote the matrix
• This matrix A has three rows and four columns and
  is said to be a 3 x 4 matrix or of dimension 3x4.
  
• We denote the element on the second row and
  fourth column with a2,4.

3
Solving a System Using Augmented Matrices
(GAUSS-JORDAN)

• Equation 1 2x 3y 2
• Equation 2 5x 4y 12

Note equation system can be written in matrix
notation as above. Coefficient matrix times
variable column vector equals vector of
constants. Also note conformability is satisfied
(2x2) (2x1) (2x1).
4
Augmented Matrix

• Augmented matrix corresponding to the system
• Our goal is to use row operations to transform
  our augmented matrix into an augmented matrix of
  the form

5
Reduced Row Echelon Form

• For an m x n matrix to be in reduced row echelon
  form, it must satisfy the following
• 1. All rows containing only zeros are at the
  bottom of the matrix
• 2. The first (leftmost) nonzero entry in a
  nonzero row is a 1. This is called the leading 1
  of its row.
• 3. The leading 1 of each nonzero row lies left of
  the leading 1 of any lower row
• 4. For any column that has a leading 1, the other
  entries in the column are zeros

6
Reduce Echelon Form

• Elementary Operations
• 1 Interchange any two rows
• 2 Multiply any row by any non zero constant
• 3 Add any multiple of any row to any other row.

7
Steps

• First step get a 1 in the upper left by using
  R1only.
• Second get a 0 in the lower left by using R1R2
  to get R2.
• Third get a 1 in the 2nd row, 2nd column by using
  R2 only.
• Fourth get a 0 in the 1st row and 2nd column by
  using R2R1to get R1
• Multiply R1 by ½

8
Continued

• Solution
• Step 2 multiply R1 by 5 and add R2 to get new R2

9
Continued

• Step 3 multiply R2 by 2/7
• Step 4 multiply R2 by 3/2 and add R1 to get R1

10
Example Solving a System Using Augmented Matrices

• 1. Structure the following system in matrix
  notation.
• Equation1 x y 2z 20
• Equation2 x 2y 30
• Equation3 x y z 20

11
Continued

• Augmented matrix

12
Continued

• Multiply R1 by (-). Add R1to R2 and R1 to R3

13
Continued

• Multiply R3 by (-). MultiplyR3 by 2 and add to
  R2. Multiply R3 by 2 and Add to R1.

14
Continued

• Multiply R2 by (-). Add R2 to R1.

15
MATRICES OPERATIONS
16
Sum or Difference of Matrices

• To add(subtract) two matrices they must be of the
  same order. This means that they must have the
  same number of rows and columns.
• To add(subtract) two such matrices, we simply
  add(subtract) the corresponding elements.

17
Sum Difference of Matrices
18
Matrix

• To multiply a matrix by a scalar means to
  multiply that matrix by a single number.
• In order to perform scalar multiplication, we
  merely multiply each element in the matrix by
  this number(ie. the scalar).

19
Example of Scalar Multiplication
20
TRANSPOSE

• If A is a m x n matrix with entries ai,k, then
  the transpose of A denoted A is a n x m matrix
  with entries ak,i,

T
21
Matrix Multiplication

• In this case we want to develop the technique for
  multiplying two matrices.
• This product is defined only if matrix A is (m x
  n) and matrix B is a (n x p).So the number of
  columns in A has to be equal to the number of
  rows in B. Matrices are said to be
  conformable.The product, C AB then is a (m x
  p) matrix.The element of the ith row and the jth
  column of the product is found by multiplying the
  ith row of A by the jth column of B. ci,j sumk
  (ai,k.bk,j)

22
Example of Matrix Multiplication
Here the of columns in A is equal to the of
rows in B
23
Example of Matrix Multiplication
Find (a) AB and (b) BA
To obtain the entries in the first row of AB,
multiply the first row 1,3 of A by the columns
of B
24
Example of Matrix Multiplication
THUS AB
25
Example of Matrix Multiplication

• B is a 2x3 and A is a 2x2. And the number of
  columns in B is not equal to the number of rows
  in A. Hence, the product BA is not defined.

26
MATRIX INVERSE AND SOLVING SYSTEMS OF LINEAR
EQUATIONS
27
Identity Matrix

• An Identity matrix is a matrix that behaves like
  the multiplicative identity 1. If I is an
  identity matrix and A is another matrix then
  IAA. An identity matrix exists only for square
  matrices. The identity matrix of dimension m x n
  has 1s along the diagonal from upper to lower
  right and 0s elsewhere.

28
Continued

• A square matrix is a matrix with the same number
  of rows as columns.
• An n x n square matrix is said to be of order n
  and is sometimes called an s-square matrix. The
  operations of addition, multiplication, scalar
  multiplication, and transpose can be performed on
  any n-square matrices, and the result is again an
  n-square matrix.

29
Invertible Matrices

• A square matrix A is said to be invertible if
  there exists a matrix B with the property that
• ABBAI, the identity matrix
• Such a matrix B is unique and it is called the
  inverse of A and is denoted by
• Note that A is the inverse of B if and only if B
  is the inverse of A.

30
Identity

• Example I x AA

31
Inverses

• Example A x BI
• Thus A and B are inverses

32
Find the Inverse of a Matrix

• To find the inverse of Matrix A
• The inverse should have a form
• If we multiply A x Inverse

33
Continued

• This equation is the same as the systems of
  linear equations
• 1a2c1 and 2a3c0
• 1b2d0 and 2b3d1
• From previous examples (SLIDE 4 Solving a System
  Using
• Augmented Matrices (GAUSS-JORDAN)) we write an
• augmented matrix

34
Continued

• First step get a 1 in the upper left by using
  R1only.
• Second get a 0 in the lower left by using R1R2
  to get R2.
• Third get a 1 in the 2nd row, 2nd column by using
  R2 only.
• Fourth get a 0 in the 1st row and 2nd column by
  using R2R1to get R1

35
Continued

• Solution multiply R1 by 2 and add to R2
• Multiply R2 by 2 and add to R1

36
Continued

• Multiply R2 by 1
• Solution

37
Example

• Solve the system of linear equations by using the
  inverse of the coefficient matrix.
• x 3y 7
• 4x 2y 9
• Write the augmented matrix

38
Continued

• First step get a 1 in the upper left by using
  R1only.
• Second get a 0 in the lower left by using R1R2
  to get R2.
• Third get a 1 in the 2nd row, 2nd column by using
  R2 only.
• Fourth get a 0 in the 1st row and 2nd column by
  using R2R1to get R1
• Solution we have a 1 in the upper left.

39
Continued

• Multiply R2 by 1/4 and add to R1
• Multiply R2 by 2/5

40
Continued

• Multiply R2 -3 and add to R1

41
Using a matrix inverse to Solve A System
bConstants
ACoefficients
XVariables
So equation system is Ax b
42
Continued

• Solve for x and y by

43
Example

• 1. Structure the following system in matrix
  notation.
• Equation1 x y 2z 20
• Equation2 x 2y 30
• Equation3 x y z 20

44
Finding an Inverse Matrix

• Finding an Inverse Matrix
• Make the augmented matrixA I
• Use the row operations to reduce A I to the
  form I A
• Write the augmented Matrix

-1
45
Continued

• Use the Gauss Jordan Method to solve.
• Multiply R1 by-1 and add R2 and R3
• Multiply R2 by 1 and add R1

46
Continued

• Multiply R3 by 2 and add R2 and multiply R3 by 4
  and add R1
• Multiply R3 by -1

47
Solve Previous Equation System
A inverse
48
Using a matrix inverse to Solve A System
bConstants
ACoefficients
XVariables
So equation system is Ax b
49
Solve Previous Equation System
A inverse
50
Continued

• If you multiply A inverse x the constants we will
  find the solution

51
Solving
A inverse
b
x
52
Solving
Note this provides the solutions for x, y, z.
Thus, x10, y10, 7 z0.
53
Leontief Input-Output Models

• Wassily Leontief created the model.
• He first published it in 1965.
• He received a Nobel Prize in 1973 for this work.
• The basic idea is that the outputs of some
  industries are the inputs of others, and you can
  keep track of this with a matrix.

54
Input-Output Models

• Capture inter-industry transactions
• Industries use the products of other industries
  to produce their own products.
• For example - automobile producers use steel,
  glass, rubber, and plastic products to produce
  automobiles.
• Outputs from one industry become inputs to
  another.
• When you buy a car, you affect the demand for
  glass, plastic, steel, etc.

55
Input-Output Models (page 123)

• Lets consider a simple model of an economy in
  which only 3 goods are produced Petroleum,
  Transportation, and chemicals. These 3 industries
  are called sectors in a 3-sector economy
• Any industry that does not produce petroleum,
  transportation, or chemicals is said to be
  outside the economy. However industries outside
  the economy may have demands on the outputs of
  these three sectors. These type of demand are
  called external demands.
• Also, the three sectors tend to use their own
  outputs. For example Transportation demand as
  input oil from Petroleum. These types of demands
  among the three sectors are referred as internal
  demands

56
Basic Input-Output Internal Demands
Chemicals
Transportation
Petroleum
Petroleum
Transportation
57
External Demands
Chemicals
Manufacturing
Agriculture
OUTSIDE THE ECONOMY
58
Determining Internal Demand

• Lets consider the following for our three
  sectors
• The production of 1 worth of petroleum (P)
  requires 0.10 of itself, 0.20 of transportation
  and 0.40 of chemicals.
• The production of 1 worth of transportation (T)
  requires 0.10 of itself, 0.60 of petroleum and
  0.25 of chemicals.
• The production of 1 worth of chemicals (C)
  requires 0.20 of itself, 0.20 of petroleum and
  0.30 of transportation .

59
Continued

• Therefore the input-output matrix T

Output
Petr
Tran
Chem
Petr
Input
Tran
Chem
60
Continued

• If the three-sector economy produces 900 million
  of P, 850 million of T and 800 million of C,
  determine how much of this production is consumed
  internally.Lets use a matrix to represent the
  information above.
• Matrix P is the Total Production matrix

61
Continued

• Lets determine how much of this production is
  consumed internally (also known as internal
  demand TxP) .
• The internal demand for P, T and C is 760m,
  505m and 732 m.

62
External Demand

• To determine the external demand matrix D, we
  simply subtract internal demand from matrix P.
• DP-TP
• Also ProductionInternal Demand External Demand
• PTPD

63
Determining Total Output

• Given the technological matrix T and the external
  demand matrix D determine the total output matrix
  P.
• PTPD
• Lets rename P with X
• XTXD or DX-TX or DX(1-T)
• when using a matrix 1 is the identity matrix.
• DX(I-T) now provided that I-T has an inverse we
  try to solve for X

64
Continued

• Solution

XTXD
X-TXD
(I-T)XD
65
Continued

• Solution

Simplification
66
Continued

• Solution

67
End Lesson 2