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									               Jim Jack (J²)

MATH 1324 – Math for Business & Economics

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Attendance
4. Systems of Equations and Inequalities
  4.1. Review: Systems of Linear Equations in
    Two Variables

DFN: Given the linear system
                    ax  by  h
                    cx  dy  k
a,b,c,d,h,k real constants,  x0 , y0  is a solution
of this system if each equation is satisfied by the
pair. The set of all such ordered pairs is called
the solution set.

Graphing

If 2 adult tickets and 1 child ticket cost $8,
while 1 adult ticket and 3 child tickets cost $9,
what is the price of each?

2x  y  8
x  3y  9
x  2y  2
x y 5




 x  2 y  4
2x  4 y  8




2x  4 y  8
 x  2y  4
DFN: A system is consistent if at least one
solution exists, otherwise, it is inconsistent. A
consistent system is independent if there is a
unique solution (exactly one solution),
otherwise it is dependent. Two systems are
equivalent if they have the same solution set.

Thm 1: The linear system
                   ax  by  h
                   cx  dy  k
must have either
A: exactly one solution
B: no solution
C: infinitely many solutions.
There are no other possibilities.
Solve a system with a graphing calculator

5 x  2 y  15
2 x  3 y  16




Method of substitution
 1. Solve one equation for one variable
 2. Sub the result into the other equation, then
    solve for the other variable.
 3. Solve equation from step 1 using step 2.

5x  y  4
2x  3y  5
Elimination by addition

Thm 2: A system of linear equations is
transformed into an equivalent system if

A: Two equations are interchanged

B: An equation is multiplied by a non-zero
constant (both sides).

C: A constant multiple of one equation is added
to another equation.

3x  2 y  8
2 x  5 y  1
2 x  6 y  3
  x  3y  2



   x 1 y  4
       2
 2x    y  8


A woman wants to use milk and orange juice to
increase the amount of calcium and vitamin A in
her daily diet. An ounce of milk contains 37 mg
of calcium and 57  g of vitamin A. An ounce of
orange juice contains 5mg of calcium and 65  g
of vitamin A. How many ounces of milk and OJ
should she drink each day to provide exactly 500
mg calcium and 1200  g vitamin A?
The quantity of a product that people are willing
to buy during some period of time depends on
its price. Generally, the higher the price, the
less demand. Similarly, the quantity of a
product that a supplier is willing to sell during
some period of time increases with the price.

Suppose the data collected on the sale of
cherries shows that:

            p  0.07q  0.76 supply
            p  0.2q  4    demand

Supply/Demand @ $2.09/lb, $1.32/lb, equilibrium
4.2 Systems of Linear Equations and Augmented
Matrices

A matrix is a rectangular array of numbers.

                                        4 5 12 
                                       0 1 8
 1  4 5                                       
 7 0  2                               3 10 9 
         
                                         6 0  1
                                                 

 a11  1    a12  4        a13  5
 a21  7 a22  0            a23  2

The dimension of a matrix is m  n if it has m
rows and n columns. A square matrix is n  n .

  a1 x  b1 y  c1 z  d1               a1 b1 c1 d1 
                                       a b c d 
 a2 x  b2 y  c2 z  d 2               2 2 2 2
 a3 x  b3 y  c3 z  d 3               a3 b3 c3 d 3 
                                                     
square matrix m  n
row matrix m  1
column matrix n  1
principal diagonal i  j

coefficient, constant, augmented matrix

2x  3y  5
 x  2 y  3
Thm 1 (like thm2 from last unit):

An augmented matrix is transformed into a row-
equivalent matrix by performing any of the
following row operations.


A: two rows are interchanged      Ri  R j
B: A row is multiplied by a non-zero constant
                    kRi  Ri
C: a constant multiple of one row is added to
another row. kR j  Ri  Ri

Gauss-Jordan elimination

1. Do row ops to get a one at the upper left.
2. Do row ops to get a zero below that.
3. Do row ops to get a one at bottom of second
   column.
4. Do row ops to get a zero above it.
3 x1  4 x2  1
 x1  2 x2  7




2 x1  3 x2  6
3x1  4 x2  1
                  2
 2 x1  x2  4
 6 x1  3 x2  12




2 x1  6 x2  3
 x1  3 x2  2
4.3 Gauss-Jordan Elimination

3 possibilities:
a11 a12 k1
               
a21 a22 k 2




Reduced Row-Echelon Form

1 0 0  1                     1 4 0 0  3
0 1 0 1          1 0 0      0 0 1 0 2 
                 0 1 4                 
                         
0 0 1  2 
                              0 0 0 1 6 
                                           


       1 0 4 5              1 0 3 
       0 1 3 0              0 1  1
                                   
       0 0 0 1 
                            0 0 0 
                                     
A matrix is said to be in reduced row echelon
form if

 1. Each row consisting entirely of zeros is
    below any row having at least one nonzero
    element.
 2. The left-most non-zero element in any row
    is a one.
 3. All other elements in the column containing
    the leftmost 1 of a given row are zeros.
 4. The leftmost 1 in any row is to the right of
    the leftmost 1 in the row above.

Main diagonal
   1s, then maybe zeros
   First non-zero element is a 1 (leading 1)
   Leftmost leading 1 listed first
   Rows with only zeros at the bottom
   All elements below main diagonal are zeros
1 0 0 1 
0 2 0 3
        
 0 0 1 5
        


0 1  2 
1 0 3 
        


1 2  2 3 
0 0 1  1
          


1 0  3 
0 0 0 
        
0 1  2 
        
2x  2 y  z  3
3x  y  z  7
 x  3y  2z  0
Gauss-Jordan Elimination

1. Choose the leftmost nonzero column, and use
   row operations to get a 1 at the top.
2. Use multiples of the row containing the 1 at the
   top to get zeros in all the remaining places in the
   column containing the 1.
3. Repeat step 1 with the submatrix formed by
   mentally deleting the row used in step 2 and all
   rows above this row.
4. Repeat step 2 with the entire matrix including
   all the rows deleted mentally. Continue until
   matrix is in reduced row echelon form.
 2 x  4 y  z  4
 4x  8 y  7z  2
 2 x  4 y  3z  5
  3x  6 y  9 z  15
 2 x  4 y  6 z  10
 2 x  3 y  4 z  6
 x1  2 x2  4 x3  x4  x5  1
2 x1  4 x2  8 x3  3 x4  4 x5  2
  x1  3 x2  7 x3       3 x5  2
A company that rents small moving trucks wants to
purchase 25 trucks with a combined capacity of
28000 cubic feet. Three different types of trucks
are available: a 10 foot truck with a capacity of 350
cubic feet, a 14 foot truck with a capacity of 700
cubic feet, and a 24 foot truck with a capacity of
1400 cubic feet. How many trucks of each type
should the company purchase?
4.4 Matrices: Basic Operations

Two matrices are equal if they are the same size
and all corresponding elements are equal ( aij  bij i, j )

 a b c  u v w a  u b  v c  w
d e f    x y z  iff d  x e  y f  z
                

Addition
The sum of two matrices of the same size is
the matrix with elements that are the sum of the
corresponding elements of the given matrices.

a b   w x  a  w b  x 
c d    y z    c  y d  z 
                            


 2  3 0   3 1 2
 1 2  5    3 2 5  
                    
              1 7
5 0  2 
            0 6 
1  3 8  
                 
             2 8 
                  

Properties of Matrices

Addition is commutative A  B  B  A

Addition is associative  A  B  C  A  B  C 

A matrix with all zero elements is the zero
matrix.

                         0 
                         0    0 0 0 0 
        0 0                 0 0 0 0 
0 0 0 
         0 0
                        0            
                         0    0 0 0 0 
                                        
                          
The negative of a matrix M, denoted  M , is a
matrix with elements that are the negatives of
the elements in matrix M.

       a b                a  b 
If M       , then  M    c  d  .
       c d                        

We can then define subtraction as
A  B  A   B.

3  2    2 2 
5 0    3 4 
              

The product of a scalar (real number) k and a
matrix M, denoted kM, is the matrix formed by
multiplying each element of M by the number k.

     3 1 0 
 2  2 1 3  
              
     0  1  2
              
Ms. Smith and Mr. Jones are salespeople in a
new car agency that sells only two models.
August was the last month for this year’s
models, and next year’s models were introduced
in September. Gross dollar sales for each month
are given in the following matrices:

       August Sales          September Sales
       Compact Luxury        Compact Luxury
Smith $54,000 $88,000        $228,000 $368,000
Jones $126,000      0        $304,000 $322,000
       (Matrix A)            (Matrix B)

What were the combined sales in August and
September for each salesperson by model?




What was the increase in dollar sales from August
to September?
      If both salespeople receive 5% commission on gross
      dollar sales, what is the commission each receives
      by model for September?



      Product of a row matrix with a column matrix
      The product of a 1 n matrix with an n 1 matrix is
      an 11 matrix given by:

                        b1 
                       b 
                        2
a1    a2   a3    an b3   a1b1  a2b2  a3b3    anbn 
                       
                        
                       bn 
                        

                 5
      2  3 0 2  
                
                 2
                
A factory produces a slalom water ski that requires
3 labor hours in the assembly department and 1
labor hour in the finishing department. Assembly
personnel receive $9/hour while finishing personnel
receive $6/hour. What is the labor cost per ski?

     9 
3 1  
     6 

Matrix product
If A is an m p matrix and B is a p n matrix, the
matrix product of A and B, denoted AB, is an m  n
matrix whose ij element is obtained by the product
of the i row of A with the j column of B. The
elements are  aip bpj . If the number of columns of
             p 1, n
A is not equal to the number of rows of B, the
matrix product is not defined.


             m p                p n
            1 3
 2 3  1 
 2 1 2   2 0 
                
             1 2
                 


 2 1
 1 0  1  1 0 1  
        2 1 2 0
                  
  1 2
      

              2 1
1  1 0 1  
 2 1 2 0    1 0 
                  
               1 2
                   


  2 6  1 2
  1  3 3 6 
            

1 2  2 6 
3 6  1  3 
           
           5
2  3 0 2  
          
           2
          

  5
 2 2  3 0 
 
  2
 

Notes: Matrix multiplication is not commutative.
The principle of zero products does not hold.

Combine the time and labor costs for slalom and
trick skis from previous example.
 Wages Hours
12 13 5 1.5
 7 8  3 1  
              
Hours Wages
5 1.5 12 13
3 1   7 8  
              
4.5 Inverse of a Square Matrix

The identity element for multiplication for the set
of all square matrices of order n is the square matrix
of order n, denoted by I, with 1's along the principal
diagonal and 0's elsewhere.

                         1 0 0 
1      0               0 1 0 
0      1                     
        
                         0 0 1 
                               

1 0 0   a b      c
0 1 0   d e      f
                  
0 0 1   g h
                 i
                     

a b     c  1 0 0 
d e     f  0 1 0 
                 
g h
        i  0 0 1 
                  
1 0   a b c 
0 1   d e f 
            

          1 0 0 
a b    c
d e        0 1 0
       f
          0 0 1
          
                
                 

IM  MI  M

Multiplicative Inverse

Let M be a square matrix of order n, and I be the
identity matrix of order n. If  a matrix M 1 
                  M 1M  MM 1  I
then M 1 is called the multiplicative inverse of M
or more simply the inverse of M.

2 3  a c 
1 2   b d 
          
1  1 1   a d   g
0 2  1 b e     h
                 
2 3 0   c f
                i
                    




1  1 1 1 0 0 
0 2  1 0 1 0
              
2 3 0 0 0 1
              
Inverse of a Square Matrix M

If M I  can be transformed by row operations into
I B , then the resulting matrix B is M 1 . However,
if we obtain all 0's in any row left of the vertical
line, then M 1 does not exist.

Find the inverse of
  4  1
  6 2 
         




Find the inverse of
 2  4
 3 6 
         
blank A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
0    1 2 3 4 5 6 7 8 91 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2
                      01234567 890123 456

19 5 3 18 5 20 0 3 15 4 5

4 3 19 3 5 0 15 5
1 1  5 18 20 3 4 0
                  


91 24 66 21 80 25 9 3 72 19 20 5

Decode
46 84 85 28 47 46 4 5 10 30 48 72 29 57 38 38 57 95

      1 1 1                          5 2  1
      
using 2 1 2
                                A1=  2  1 0 
                                                
      2 3 1 
                                     4  1  1
                                                
4.6 Matrix Equations and Systems of Linear Equations

  ax  b      ax  b  c

  Basic Properties of Matrices
  Addition

   A  B   C  A  B  C 
  A B  B  A
  A0  0 A
  A   A   A  A

  Multiplication

   ABC  ABC 
  AI  IA  A
  AA1  A1 A  I

  Combined
  AB  C   AB  AC
  B  C A  BA  CA
  A  B  CA  CB and AC  BC
Solving a matrix equation

AX=B




x  y  z 1
    2y  z 1
2x  3y   1

      3    3  1
A1   2  2 1 
                
       4  5 2 
                
 x yz 3
    2y  z 1
2x  3y    4




 x  y  z  5
    2y  z  2
2x  3y     3


If the number of equations in a system equals the
number of variables, and the coefficient matrix has
an inverse, then the system will have a unique
solution that can be found using the inverse.

Will not work if:

1) coefficient matrix is singular
2) number of variables does not equal the number of
equations.
An investment advisor currently has two types of
investments available for clients: A conservative
investment A that pays 10%/year, and a higher risk
investment B that pays 20% per year. Clients may
divide investments between the two to achieve any
total return between 10% and 20%. Higher return is
higher risk. How should each client listed invest to
achieve the desired return?

                  Client
             1      2      3    k
Investment $20000 $50000 $10000 k1
   Rturn        $2400   $7500    $1300    k2
                 12%     15%      13%

   x1      x2  k1
0.1x1  0.2 x2  k 2
4.7 Leontief Input-Output Analysis

Wassily Leontief 1973 Nobel Prize
500 sectors of the economy interacting

Two industry model Electric and Water

Suppose production of $1 worth of electricity
requires $0.30 worth of electricity and $0.10 worth
of water. Production of $1 worth of water requires
$0.20 worth of electricity and $0.40 worth of water.
Outside demand is:

   d1  $12 million for electricity
   d 2  $8 million for water

To produce this amount of water and electricity:
Basic Input-Output problem
Given the internal demands for each industry’s
output, determine the output levels for the various
industries that will meet the given final (outside)
level demand as well as the internal demand.

x1  0.3 x1  0.2 x2  d1
x2  0.1x1  0.4 x2  d 2

 x1  0.3 0.2  x1   d1 
 x   0.1 0.4  x   d 
 2            2   2 

X  MX  D




       0.7  0.2                  1 .5 0 .5 
                  and  I  M   0.25 1.75
                                1
I M 
       0.1 0.6                             
Solution to a Two-Industry Input-Output Problem

Given two industries C1 and C2 , with

          C1 C2
                         x1          d1 
M  C1    a11 a12 , X    , and D   
                         x2         d 2 
    C2    a21 a22

where aij is the input required of Ci to produce a
dollar’s worth of output for C j , the solution to
the input-output matrix equation
                   X  MX  D
is
                 X  I  M  D
                              1


assuming that I  M  has an inverse.
Three-Industry model

An economy is based on three sectors, agriculture,
energy, and manufacturing. Production of a
dollar’s worth of agriculture requires an input of
$0.20 from the agriculture sector, and $0.40 from
the energy sector. Production of a dollar’s worth
of energy requires an input of $0.20 from the
energy sector, and $0.40 from the manufacturing
sector. Production of a dollar’s worth of
manufacturing requires an input of $0.10 from the
agriculture sector, $0.10 from the energy sector,
and $0.30 from the manufacturing sector. Find
the output from each sector that is needed to
satisfy a final demand of $20billion for
agriculture, $10billion from energy, and
$30billion from manufacturing.
                 A      E      M
        A

       E

       M
I M 



I  M  
      1
5. Linear Inequalities and Linear Programming
  5.1. Linear Inequalities in Two Variables

y  2x  3       2x  3y  5

y  2x  3       2x  3y  5

3 x  4 y  24   3 x  4 y  24    3 x  4 y  24
Graphing of Linear Inequalities        ax  by  c

The graph of the linear inequality Ax  By  C or
 Ax  By  C is either the upper half-plane or the
lower half-plane determined by the line Ax  By  C .

Procedure:
1) First graph Ax  By  C as a dashed line if
   equality is not included in the original
   statement or a solid line if equality is included.
2) Choose a test point anywhere in the plane not
   on the line and substitute into the inequality.
3) The graph of the original inequality includes
   the half-plane containing the test point if the
   inequality is satisfied by that point, or the
   other half-plane if the inequality is not
   satisfied.

2x  3y  6
y  3
2x  5
x  3y




Interpreting a graph

Ax  By  C
A concert sales promoter wants to book a rock
group for a stadium concert. A ticket for the
playing field costs $125, while a ticket in the
stands costs $175. The group wants a total ticket
sales of at least $700,000. How many tickets of
each type must be sold?
5.2 Systems of Linear Inequalities
x y 6
2x  y  0
Solve Systems of Linear Inequalities; Corner Pts
 2 x  y  22
   x  y  13
2 x  5 y  50
       x0
       y0
A patient in a hospital requires at least 84 units of
drug A and 120 units of drug B. Each gram of M
contains 10 units of drug A and 8 units of drug B.
Each gram of N contains 2 units of drug A and 4
units of drug B. How many grams of M and N
can be mixed to meet the minimum daily rqmts?

  M      N    MDR
A 10     2     84
B 8      4    120
5.3 Linear Programming in Two Dimensions:
Geometric Approach

Objective – find values of decision variables
that produce optimal value

Objective function maximized or minimized to
make that decision

Constraints – limits imposed in the decision

Feasible solutions (feasible region) – solution
set for system of linear equations.

Optimal solution – the best feasible solution

Iso-profit line – lines of constant profit
A tent manufacturer makes a standard model and an
expedition model. Each standard tent requires 1
labor-hour from the cutting department and 3 labor-
hours from the assembly department. Each
expedition tent requires 2 labor-hours from the
cutting department and 4 labor-hours from the
assembly department. The maximum labor hours
available per day in the cutting and assembly
departments are 32 and 84 respectively. If the
company makes a $50 profit on each standard tent,
and $80 on each expedition tent, how many tents
should be made daily to maximize profit?

     Labor hours req per tent
        Std Exp Max
   Cut     1     2    32
  Assy      3    4    84
  Profit   50   80
Maximize P  50x1  80x2
         x1  2 x2  32
             3x1  4 x2  84
Subject to
             x1  0
             x2  0
Fundamental Theorem of Linear Programming

If an optimal value for a linear programming
problem exists, then it occurs at one or more
corner points of the region of feasible solutions.

Procedure
1. Identify the decision variables.
2. Summarize variable information in a table.
3. Determine objective and linear obj function.
4. Write problem constraints (& non-neg).
5. Graphically determine feasible region.
6. Determine the corner points of feasible region.
7. Determine optimal solution by evaluating
   objective function at corner points.
8. Interpret results in terms of original problem.

Existence of feasible solutions
1. bounded region
2. unbounded region
3. empty feasible region
Minimize and maximize z  3x1  x2 , subject to
x1  0 , x2  0 , 2 x1  x2  20, 10x1  x2  36 , and
2 x1  5 x2  36




Solving a Linear Programming Problem
 1. Read problem carefully.
 2. Use a table to write the objective function
     and all of the constraints.
 3. Sketch a graph of the region of feasible
     solutions. Identify vertices.
 4. Evaluate the objective function at each one
     of the vertices. Choose the min/max.
Minimize and maximize z  10x1  20 x2 , subject
to x1  0 , x2  0 , 6 x1  2 x2  36, 2 x1  4 x2  32,
and x2  20
Minimize and maximize P  2 x1  3x2 , subject
to x1  0 , x2  0 , x1  x2  8, x1  2 x2  8, and
2 x1  x2  10
A patient in a hospital requires at least 84 units
of drug A and 120 units of drug B. Each gram
of M contains 10 units of drug A and 8 units of
drug B. Each gram of N contains 2 units of
drug A and 4 units of drug B. Now suppose that
both M and N contain an undesirable drug D, 3
units/gram in M and 1 unit/gram in N. How
many grams of M and N can be mixed to meet
the minimum daily requirements, while
minimizing intake of drug D?

    M    N    MDR
A 10      2    84
B 8       4   120
D 3       1
 6. Linear Programming: The Simplex Method
 6.1 Geometric Introduction of the Simplex Method

 Maximize P  50x1  80x2
          x1  2 x2  32
              3x1  4 x2  84
 Subject to
              x1  0
              x2  0

 Bounded and unbounded variables

Standard Form
A linear programming problem is said to be a
standard maximization problem in standard form
if its mathematical model is of the following form:
           Maximize the objective function
                 P  c1 x1  c2 x2    cn xn
   subject to constraints of the following form:
            a1 x1  a2 x2    an xn  b, b  0
   with non-negative constraints
                        x1, x2 , xn  0
Slack variables
x1  2 x2  32               x1  2 x2  s1        32
3 x1  4 x2  84            3 x1  4 x2        s2  84

Basic/Non-basic Variables
Basic Solutions
Basic Feasible Solutions



Given a system of linear equations associated
with a linear programming problem:
   The variables are divided into two (mutually
   exclusive) groups: Basic variables are
   selected arbitrarily with the restriction that
   there are as many basic variables as
   equations. The rest of the variables are
   nonbasic variables.

    A solution found by setting nonbasic
    variables to zero and solving for the basics is
    a basic solution. If a basic solution has no
    negative values, it is a basic feasible solution.
Suppose there is a system of three problem
constraint equations with eight variables
associated with a standard maximization problem.

Basic/Nonbasic variables?

Setting nonbasic variables to zero

Basic solution has all non-negative elements
   Feasible?

Fundamental Theorem of Linear Programming

If an optimal value of the objective function in
a linear programming problem exists, then it
occurs at one or more of the basic feasible
solutions.
6.2 Simplex Method: Maximization with
problem constraints of the form 

Maximize P  50x1  80x2
         x1  2 x2  32
             3x1  4 x2  84
Subject to
             x1  0
             x2  0

     x1  2 x2  s1             32
   3 x1  4 x2         s2      84
 50 x1  80 x2              P0
         x1, x2 , s1 , s2  0
Basic Solutions and Basic Feasible Solutions (bfs)
  1. Let P always be a basic variable.
  2. Note that a basic solution of this system is
     still basic after P is deleted.
  3. If a basic solution to the above is a bfs
     without P, it is also a bfs with P.
  4. A bfs can contain a negative number only
     if it is the value of P.
Fundamental Theorem of Linear Programming

If an optimal value of the objective function in
a linear programming problem exists, then it
occurs at one or more of the basic feasible
solutions of the initial system.

     x1  2 x2  s1             32
   3 x1  4 x2         s2      84
 50 x1  80 x2              P0

Initial basic feasible solution, pick slacks and P

Simplex Tableau

    x1      x2    s1 s2       P
s1  1   2 1 0 0 32
s2  3   4 0 1 0 84
                    
P  50  80 0 0 1 0 
                    
Pivot Operation
    x1    x2 s1     s2   P
s1  1   2 1 0 0 32
s2  3   4 0 1 0 84
                    
P  50  80 0 0 1 0 
                    

Pivot Element
1.Select most negative indicator in bottom row.
  This is the pivot column.
2.Divide each positive element in the pivot
  column into the corresponding element in the
  last column. (if no positive elements in pivot
  column, stop) The pivot row is the one that
  generates the smallest quotient.
3.The pivot element is at the intersection of
  pivot row and column.

Pivot operation
1. Use division to put a 1 in the pivot element.
2. Use row operations to zero all other elements
   in that column .
  x1   x2   s1 s2   P
s1  1   2 1 0 0 32
s2  3   4 0 1 0 84
                    
P  50  80 0 0 1 0 
                    
Maximize P  10 x1  5 x2
         4 x1  x2  28
             2 x1  3 x2  24
Subject to
                     x1  0
                    x2  0
Maximize P  6 x1  3x2
          2 x1  3x2  9
              x1  3x2  12
Subject to
                    x1  0
                    x2  0
A farmer owns a 100-acre farm and plans to plant
at most 3 crops. The seed for crops A, B, and C
cost $40, $20, and $30 per acre respectively. A
max of $3200 can be spent on seed. These crops
require 1, 2, and 1 workday per acre, and there are
a max of 160 workdays available. If the farmer
can realize profit of $100, $300, and $200 per acre
respectively, how many acres of each crop should
be planted to max profit?

Objective:
Constraints:
6.3 Systems of Linear Equations in Three
Variables
(end)
2x  3y  4z  4
     y  2z  0         3,2,1
 x  5 y  6z  7




 x  4 y  2 z  15
                          1,3,2
4x  y  z  1
                         1,10,13
6 x  2 y  3z  6
Solving with elimination and substitution

1. Eliminate one variable from two of the
   equations.
2. Apply the techniques from 6.1 and 6.2 to
   solve the resulting two equations.
3. Substitute back in to find the third variable.

  x  y  2z  6
2 x  y  2 z  3
 x  2 y  3z  7
One thousand tickets were sold for a play,
which generated $3800. The prices of the
tickets were $3 for children, $4 for students, and
$5 for adults. There were 100 fewer student
tickets sold than adult tickets. Find the number
of each ticket sold.
Three students buy lunch in the cafeteria. One
student buys 2 hamburgers, 1 order of fries, and a
soda for $9. Another student buys 1 hamburger,
2 orders of fries, and a soda for $8. A third
student buys 3 hamburgers, 3 fries, and 2 sodas
for $18. If possible, find the cost of each item.
x  y  z  2
x  2 y  2 z  3
     y  z  1
6.4 Solutions of Linear Systems Using Matrices

Carl Friedrich Gauss (1777-1855)
Gaussian Elimination with Backward Substitution

Matrix – a rectangular array of numbers
The dimension of a matrix is m  n if it has m
rows and n columns. A square matrix is n  n .

 a1 x  b1 y  c1 z  d1     a1 b1 c1     d1
a2 x  b2 y  c2 z  d 2     a1 b1 c1     d1
a3 x  b3 y  c3 z  d 3     a1 b1 c1     d1


3x  4 y  6
 5 x  y  5


2 x  5 y  6 z  3
3x  7 y  3z  8
 x  7y         5
1 0 2 3
2 2 10 3
1 2 3 5




1 2 3 4
0 1 6 7
0 0 1  8
Row-Echelon Form

1 3 0 1                        1 3 1 5
                   1 2 0
0 1 6 1                        0 1 1 3
                   0 1 4
0 0 1 2                        0 0 1 0


       1 3 1 5
                            1 3 5
       0 0 1 3
                            0 0 1
       0 0 0 0

Main diagonal
   1s, then maybe zeros
   First non-zero element is a 1 (leading 1)
   Leftmost leading 1 listed first
   Rows with only zeros at the bottom
   All elements below main diagonal are zeros
Solve the system

1 1 3 12
0 1 2 4
0 0 1  3




1 1 5 5
0 1 3 3
0 0 0 0
Gaussian Elimination

Matrix Row Transformations
   1. Any two rows may be interchanged
   2. The elements of any row may be
      multiplied by a non-zero constant
   3. Any row may be changed by adding or
      subtracting a multiple of another row.

  x  y  z 1
x y z 5
     y  2z  5
2x  4 y  4z  4
  x  3y  z  4
 x  3 y  2 z  1
Reduced Row-Echelon Form
Row-Echelon Form with elements above and
below the leading 1 are zeros

                    1 0 0
1 0        1 0                   1 0       3
                    0 1 0
0 1        0 0                   0 1 2
                    0 0 1

   1 0 0 3             1 0 4 8
   0 1 0 1             0 1 1 2
   0 0 1 1            0 0 0 0

Solve the system
1 0 0 3
0 1 0 1
0 0 1 2


1 0    6
0 1 5
Transform to Reduced Row-Echelon Form

2 x  y  2 z  10
 x       2z  5
 x  2 y  2z  1
6.5 Properties and Applications of Matrices
                                   a11    a12    a13    a14
              a11    a12    a13                                    a11    a12
a11    a12                         a 21   a 22   a 23   a 24                        a11   a12   a13
              a 21   a 22   a 23                                   a 21   a 22
a 21   a 22                        a 31   a 32   a 33   a 34                        a21   a22   a23
              a31    a32    a33                                    a 31   a 32
                                   a 41   a 42   a 43   a 44



An element of matrix A is designated                                         aij   ( arc )

Two matrices A and B are equal if all corresponding
elements are equal ( aij  bij i, j )

              3 3             7                    3 x        7
Let A= 1             6       2             B= 1           6 2
              4      2         5                    4 5        2


a12                     b32               a13 



What value of x would make A=B?


a31b13  a32b23  a33b33 
Sum, Difference, Scalar Multiplication

The sum of two m  n matrices A and B is the
m  n matrix A B , where each element is the
sum of corresponding elements in A and B.
aij  bij i, j  1  i  m,1  j  n

2 2 2     1 1 1
0 2 0  1 1 1
0 2 0     1 1 1

1 2 1     1 1 1
2 2 2     1 1    1
1 2 1     1 1 1


The difference of two m  n matrices A and B is
the m  n matrix A  B , where each element is
the sum of corresponding elements in A and B.
aij  bij i, j  1  i  m,1  j  n

3 3 3     1 1 1
1 3 1  1 1 1
1 3 1     1 1 1
   7  8 1             5  2 10
A=               B=
   0 1 6             3   2   4



A+B=


B+A=


A – B=




1 1 1    1 1 1
1 1 1  1 1 1
1 1 1    1 1 1



   1 1 1
2  1 1 1
   1 1 1
The product of a scalar (real number) k and
m  n matrix A is the m  n matrix kA , where
each element is the sum of corresponding
elements in A and B.
kaij i, j  1  i  m,1  j  n


     2 7      11
If A= 1   3    5,   find  4 A
     0 9  12
                               1 1         1  3
     4 2          0 1
A=           B=          C=    0   7   D=    9 7
     3   5        2 3
                              4   2         1   8



A  3B 




AC 



 2C  3D 
Matrix Products

          A B                 $
Student1 10 7        CollegeA 60
Student2 11 4        CollegeB 80


Cost for Student 1




Cost for Student 2




   10 7 60
AB=    
   11 4 80
The product of an m  n matrix with an n k
matrix is an m k matrix. The elements are
 aij b jk .
 j 1, n



                m n              n k


     1 1                                      1 1    2
                    1        1 2 3
A= 0       3   B=        C=              D=    0   3 2
                    2        4 5 6
    4      2                                  3   4   5


AB=




CA=
DC=




CD=




      1   0   7          4 6    7
A=    3   2 1        B= 8   9 10
     5 2    5          0   1   3




AB=




Graphing Calculator          2 A  3B 3
6.6 Inverses of Matrices        A1 like f 1

Translation
    1 0 h             x
A0 1 k             Xy         AX 
  0 0 1               1


Translate the point  1,2, 3 units right, down 4.




      1 0 3
A1  0 1 4
      0 0 1
AA1 




A1 A 




The n  n identity matrix, denoted by I n , has
only 1s on the main diagonal and 0s elsewhere.
Let A be an n  n matrix. If there exists an n  n
matrix, denoted A1, that satisfies

          A1 A  I n        and   AA1  I n

then A1is the inverse of A.


If A1 exists, A is invertible or non-singular

If A is not invertible, then A is singular.

           5      3           2    3
Let A                  B                B  A1?
          3 2              3 5
90° clockwise rotation
         0 1 0                  0 1 0
Let A   1 0 0 and       A1  1 0 0
         0 0     1              0 0 1

Rotate the point  2,0 clockwise 90° about origin




What will A1 do?
Find A1.
Representing linear systems with matrix equations

3x  2 y  4 z  5
2 x  y  3z  9
 x  5 y  2z  5




3x  4 y  7
 x  6 y  3



   x  5y        2
 3x  2 y  z  7
  4 x  5 y  6 z  10
Solving linear systems with inverses
AX  B




 x  4y  3                            9 4
2x  9 y  5                           2   1
 x  3y  z  6              1.25 2 .25
    2 y  z  2             .25  1  .25
 x  y  3z  4              .5  1  .5




Modeling blood pressure   P A,W   a  bA  cW
 P A W
113 39 142
138 53 181
152 65 191
Finding Inverses Symbolically

      1        1 4
Find A if A 
                2 9




                1 0 1
Find A1 if A  2 1 3
               1 1 1

								
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