W&C - Chapter 2

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

W&C - Chapter 2W&C - Chapter 2

MatricesMatrices


Using Matrices to Solve Systems of EquationsUsing Matrices to Solve Systems of Equations

1.1. MatricesMatrices are are arrays of numbersarrays of numbers that can represent many that can represent many things and have many uses.things and have many uses.

2.2. Matrices can be used to Matrices can be used to representrepresent systems of equationssystems of equations and to and to solve themsolve them systematicallysystematically..

1 10 10001 10 1000– – 1 20 5001 20 500

3.3. The horse in the previous PowerPoint really liked this The horse in the previous PowerPoint really liked this approach.approach.

I’m lazy. Tell me more!


The Coefficient Row of an EquationThe Coefficient Row of an Equation

Consider the following linear equation:Consider the following linear equation:

22xx – – yy = 3 = 3 Notice that the equation is entirely defined by its Notice that the equation is entirely defined by its coefficients coefficients

((22 and and 11) and by its ) and by its constant termconstant term ( (33).).





((22 and and 11) and by its ) and by its constant termconstant term ( (33).). If we were simply given the row of numbers If we were simply given the row of numbers

[2 – 1 3][2 – 1 3]

we could easily reconstruct the original linear equation as we could easily reconstruct the original linear equation as follows:follows:

(2) (2) xx + (– 1) + (– 1) yy = (3) = (3)

22xx – – yy = 3 = 3





((22 and and 11) and by its ) and by its constant termconstant term ( (33).). If we were simply given the row of numbers If we were simply given the row of numbers

[2 – 1 3][2 – 1 3]

we could easily reconstruct the original linear equation as we could easily reconstruct the original linear equation as follows:follows:

(2) (2) xx + (– 1) + (– 1) yy = (3) = (3)

22xx – – yy = 3 = 3 We call such a row the We call such a row the coefficient rowcoefficient row of an equation. of an equation.


Multiplying The Coefficient RowsMultiplying The Coefficient Rows

Multiplying both sides of an equation by a number Multiplying both sides of an equation by a number corresponds to multiplying the coefficient row by the same corresponds to multiplying the coefficient row by the same number.number.

Example:Example:

EquationEquation RowRow

EE11:: 22xx – – yy = 3 = 3

[ 2 – 1 3 ][ 2 – 1 3 ] RR11

Multiply byMultiply by – 2: – 2: (– 2) (– 2) EE11:: – – 44xx + 2 + 2yy = – 6 = – 6 [ – 4 2 – 6 ][ – 4 2 – 6 ] (– 2)(– 2) RR11


Adding The Coefficient RowsAdding The Coefficient Rows

Adding two equations corresponds to adding their Adding two equations corresponds to adding their coefficient rows.coefficient rows.

Example:Example:

EquationEquation RowRow

EE11:: 22xx – – yy = 3 = 3 [ 2 – 1 3 ][ 2 – 1 3 ] RR11

EE22:: – – xx + 2 + 2yy = – 4 = – 4

[ – 1 2 – 4 ][ – 1 2 – 4 ] RR22

Add:Add: EE11 + + EE22:: xx + + yy = – 1 = – 1 [ 1 1 – 1 ][ 1 1 – 1 ] RR11 + + RR22


Elementary Row OperationsElementary Row Operations

There are three types of row operations:There are three types of row operations: Type 1:Type 1: Replacing Replacing RRii by by aRaRii (where (where aa = 0 = 0), also ), also

known as “multiplying”known as “multiplying” Example:Example:

3R2

1 3 4

0 4 2



There are three types of row operations:There are three types of row operations: Type 1:Type 1: Replacing Replacing RRii by by aRaRii (where (where aa = 0 = 0))

Example:Example:

1 3 4

0 4 2

3R2

1 3 4

0 12 6



There are three types of row operations:There are three types of row operations: Type 2:Type 2: Replacing Replacing RRii by by aRaRii + + bRbRjj

Example:Example:

1 3 4

0 4 2

4R1 – 3R2



There are three types of row operations:There are three types of row operations: Type 2:Type 2: Replacing Replacing RRii by by aRaRii + + bRbRj j

Also known as “adding a multiple of one row to another”Also known as “adding a multiple of one row to another” Example:Example:

1 3 4

0 4 2

4R1 – 3R24 0 22

0 4 2

This example is about as complicated as it gets. This example is about as complicated as it gets. Watch the slow demonstration on the chalkboardWatch the slow demonstration on the chalkboard



There are three types of row operations:There are three types of row operations: Type 3:Type 3: Switching the order of the rows.Switching the order of the rows.

Example:Example:

1 3 4

0 4 2

1 2 3

R1 R2



There are three types of row operations:There are three types of row operations: Type 3:Type 3: Switching the order of the rows.Switching the order of the rows.

Example:Example:

1 3 4

0 4 2

1 2 3

R1 R2

0 4 2

1 3 4

1 2 3

It’s important to note that It’s important to note that row operationsrow operations do not change the do not change the solutionssolutions of the corresponding systems of equations: of the corresponding systems of equations: The new system of equations that we get by applying these The new system of equations that we get by applying these

row operations has the row operations has the same solutionssame solutions as the original one. as the original one.

You can use You can use ExcelExcel to perform to perform row operationsrow operations. . The text shows you how to do this on The text shows you how to do this on W&C pages W&C pages

99 and 10099 and 100, but , but don’t do itdon’t do it..


Solving Systems of Equations Using Row OperationsSolving Systems of Equations Using Row Operations

Consider this system of equations:Consider this system of equations:

23

3 211

4 4

x y

xy



Consider this system of equations:Consider this system of equations:

This system corresponds to the following This system corresponds to the following augmented matrixaugmented matrix::

23

3 211

4 4

x y

xy

2 1

3 21 11

4 4

3

1

We will use this matrix to solve the system of equations.We will use this matrix to solve the system of equations.



Step 1:Step 1: Fractions are messy and annoying. Fractions are messy and annoying. Clear the fractionsClear the fractions and/or decimals (if any) and/or decimals (if any) using operations of type 1:using operations of type 1:

2 1

3 21 11

4 4

3

1

6R1

4R2

4 3 18

1 4 11



Step 2:Step 2: Designate the first nonzero entry in the first row as Designate the first nonzero entry in the first row as the the pivotpivot::

4 3 18

1 4 11

Pivot column

Pivot row



4 3 18

1 4 11

Pivot column

Pivot row

Step 3:Step 3: Clear the pivot columnClear the pivot column using operations of using operations of type 2type 2.. ““Clearing the column” means changing the matrix so that Clearing the column” means changing the matrix so that

the pivot is the only nonzero number in the column.the pivot is the only nonzero number in the column.

4R2 1R1

Focusing on the pivot Focusing on the pivot column, column, multiplymultiply each each row by the row by the absolute valueabsolute value of the entry currently in of the entry currently in the other.the other.



4 3 18

1 4 11

Pivot column

Pivot row

Step 3:Step 3: Use the pivot to Use the pivot to clear the pivot columnclear the pivot column using using operations of operations of type 2type 2.. ““Clearing the column” means changing the matrix so that Clearing the column” means changing the matrix so that


4R2 1R1+

If the entries in the If the entries in the pivot column have pivot column have opposite signsopposite signs, insert a , insert a plus (plus (++).).

Otherwise insert a Otherwise insert a negative (negative (––).).



4 3 18

1 4 11

Pivot column

Pivot row

Step 3:Step 3: Use the pivot to Use the pivot to clear the pivot columnclear the pivot column using using operations of operations of type 2type 2.. ““Clearing the column” means changing the matrix so that Clearing the column” means changing the matrix so that


4R2 1R1+

4 3 18

0 13 26



Optional:Optional: if all the numbers in a row are multiples of an if all the numbers in a row are multiples of an integer, divide by that integer to integer, divide by that integer to simplifysimplify::

4 3 18

0 13 26

2

1

13R

4 3 18

0 1 2


1R1 3R2


Step 4:Step 4: Select the first nonzero number in the Select the first nonzero number in the second rowsecond row as as the the pivotpivot and and clear its columnclear its column::

4 3 18

0 1 2

Pivot column

Pivot row

+ 4 0 12

0 1 2



Step 5:Step 5: (Final Step) Using operations of type 1, (Final Step) Using operations of type 1, turn each turn each pivot into a 1pivot into a 1::

1

2

1

4R

R

4 0 12

0 1 2

1 0 3

0 1 2

Turning this Turning this reduced matrixreduced matrix back into a back into a system of equationssystem of equations we find we find the solutionthe solution to the original system of equations: to the original system of equations:

1 0 3

0 1 2

x y

x y

3

2

x

y

or:or:

This procedure is called the This procedure is called the Gauss-Jordan reductionGauss-Jordan reduction or or row row reductionreduction..


Using Matrices to Solve a System of Three EquationsUsing Matrices to Solve a System of Three Equations

Solve the system using the Solve the system using the Gauss-Jordan reductionGauss-Jordan reduction method: method:

5 6

3 3 10

3 2 5

x y z

x y z

x y z



Convert the system of equations into an Convert the system of equations into an augmented matrixaugmented matrix::

Note that there are Note that there are no fractions or decimalsno fractions or decimals to be cleared in to be cleared in this system.this system.

5 6

3 3 10

3 2 5

x y z

x y z

x y z

1 1 5 6

3 3 1 10

1 3 2 5


1 1 5 6

3 3 1 10

1 3 2 5


Use the pivot to Use the pivot to clear the clear the firstfirst column column using operations using operations of of type 2type 2..


1 1 5 6

3 3 1 10

1 3 2 5


Pivot column

Pivot row


1R2 3R1–

1R3 1R1–

1 1 5 6

0 6 16 28

0 4 3 11


1 1 5 6

0 6 16 28

0 4 3 11


Simplify:Simplify:

21

2R

1 1 5 6

0 3 8 14

0 4 3 11



Use the pivot of the Use the pivot of the secondsecond row row to to clear the clear the secondsecond column column using operations of using operations of type 2type 2::

1 1 5 6

0 3 8 14

0 4 3 11


1 1 5 6

0 3 8 14

0 4 3 11


Pivot column

Pivot row


3R1 1R2+

3R3 4R2–

3 0 7 4

0 3 8 14

0 0 23 23



Simplify:Simplify:

31

23R

3 0 7 4

0 3 8 14

0 0 23 23

3 0 7 4

0 3 8 14

0 0 1 1



Use the pivot of the Use the pivot of the thirdthird row row to to clear the clear the thirdthird column column using operations of using operations of type 2type 2::

3 0 7 4

0 3 8 14

0 0 1 1


3 0 7 4

0 3 8 14

0 0 1 1


Pivot column

Pivot row


1R1 7R3–

1R2 8R3+

3 0 0 3

0 3 0 6

0 0 1 1


3 0 0 3

0 3 0 6

0 0 1 1


Using operations of type 1, Using operations of type 1, turn each pivot into a 1turn each pivot into a 1::

1 0 0 1

0 1 0 2

0 0 1 1

1

2

1

31

3

R

R

Transform matrix back into a Transform matrix back into a system of equationssystem of equations and find the and find the solutionsolution::

1 0 0 1

0 1 0 2

0 0 1 1

x y z

x y z

x y z

or:

1

2

1

x

y

z

Check the solutionCheck the solution by substituting in the original system. by substituting in the original system.

W&C pages 105-106 shows you how to use Excel for row reduction,

but don’t bother yet.


The Main Diagonal The Main Diagonal

The goal in the Gauss-Jordan reduction is to reduce the The goal in the Gauss-Jordan reduction is to reduce the matrix to the following form:matrix to the following form:

This method aims to place This method aims to place 1s1s into all the elements in the into all the elements in the main diagonalmain diagonal, with all , with all 0s0s above and below. above and below.

Once the matrix has this form, returning it to a Once the matrix has this form, returning it to a system of system of equationsequations yields the yields the solutionsolution..

1 0 0 #

0 1 0 #

0 0 1 #


Example: Example: Using Gauss-Jordan ReductionUsing Gauss-Jordan Reduction Bring out a blank sheet of scrap paper and use it to solve the Bring out a blank sheet of scrap paper and use it to solve the

system:system:

Step 0: Step 0: Convert the system of equations into an Convert the system of equations into an augmented matrixaugmented matrix..Step 1:Step 1: Clear any fractions and/or decimals using operations of type 1. Clear any fractions and/or decimals using operations of type 1.Step 2:Step 2: Designate the first nonzero entry in the first row as the Designate the first nonzero entry in the first row as the pivotpivot..Step 3:Step 3: Clear the pivot columnClear the pivot column using operations of type 2. using operations of type 2.Step 4:Step 4: Select the first nonzero number in the Select the first nonzero number in the second rowsecond row as the as the pivotpivot and and

clear its columnclear its column. . (Do the same for the following rows, if any)(Do the same for the following rows, if any)Step 5:Step 5: Using operations of type 1, Using operations of type 1, turn each pivot into a 1turn each pivot into a 1..Optional at any point:Optional at any point: if all the numbers in a row are multiples of an if all the numbers in a row are multiples of an integer, divide by that integer to integer, divide by that integer to simplifysimplify..

2 3 1

4 2 4 4

2 4

x y z

x y z

x y z


Example: Example: Using Gauss-Jordan ReductionUsing Gauss-Jordan Reduction

Convert the system of equations into an Convert the system of equations into an augmented matrixaugmented matrix::

2 1 3 1

4 2 4 4

1 2 1 4

2 3 1

4 2 4 4

2 4

x y z

x y z

x y z

There are There are no fractions or decimalsno fractions or decimals to be cleared in this system, but to be cleared in this system, but the second row can be the second row can be simplifiedsimplified::

2 1 3 1

4 2 4 4

1 2 1 4

21

2R

2 1 3 1

2 1 2 2

1 2 1 4


2 1 3 1

2 1 2 2

1 2 1 4


Pivot column

Pivot row


2R2 2R1–

2R3 1R1–

2 1 3 1

0 0 2 2

0 3 1 7



Switch rows two and threeSwitch rows two and three to ensure the main diagonal to ensure the main diagonal remains nonzero:remains nonzero:

2 1 3 1

0 0 2 2

0 3 1 7

R2 R3

2 1 3 1

0 3 1 7

0 0 2 2




2 1 3 1

0 3 1 7

0 0 2 2


2 1 3 1

0 3 1 7

0 0 2 2


Pivot column

Pivot row


3R1 1R2–

R3

6 0 10 4

0 3 1 7

0 0 2 2




6 0 10 4

0 3 1 7

0 0 2 2


6 0 10 4

0 3 1 7

0 0 2 2


Pivot column

Pivot row


2R1 10R3+

2R2 1R3

12 0 0 12

0 6 0 12

0 0 2 2

–


12 0 0 12

0 6 0 12

0 0 2 2


Using operations of type 1, Using operations of type 1, turn each pivot into a 1turn each pivot into a 1::

1 0 0 1

0 1 0 2

0 0 1 1

1

2

3

1

121

61

2

R

R

R

Transform matrix back into a Transform matrix back into a system of equationssystem of equations and find the and find the solutionsolution::

1 0 0 1

0 1 0 2

0 0 1 1

x y z

x y z

x y z

or:

1

2

1

x

y

z

Check the solutionCheck the solution by substituting in the original system. by substituting in the original system.


Example: Example: An Inconsistent SystemAn Inconsistent System

Solve the system:Solve the system:

1

2 0

4 3 3

x y z

x y z

x y z


1 1 1 1

0 3 1 2

0 3 1 1

Example: Example: An Inconsistent SystemAn Inconsistent System

Pivot column

Pivot row


3 0 2 1

0 3 1 2

0 0 0 1

3R1 R2+

R3 R2–

The last row translates into The last row translates into 0 = 10 = 1, which is nonsense., which is nonsense. This system has This system has no solutionno solution: :

There are no numbers for There are no numbers for xx, , yy and and zz that will lead to that will lead to 0 = 10 = 1.. We say that the system is We say that the system is inconsistentinconsistent. .


Example: Example: Infinitely Many SolutionsInfinitely Many Solutions

Solve the system:Solve the system:

31 14 2 4

1

0

7 3 3

x y z

x y z

x y z



There are There are no nonzero entriesno nonzero entries in the third row, so there can in the third row, so there can be be no pivotno pivot in the third row. in the third row.

Se we Se we skip to the final stepskip to the final step by turning the pivots into 1s: by turning the pivots into 1s:

3 0 5 2

0 3 2 1

0 0 0 0

5 23 3

2 13 3

1 0

0 1

0 0 0 0

113

123

R

R



TranslatingTranslating back into equations back into equations we obtain: we obtain:5 23 3

2 13 3

x z

y z

523 3

1 23 3

x z

y z

Solving for Solving for xx and and yy we get the we get the solutionsolution::

You can choose You can choose any value you likeany value you like for for zz.. For any value ofFor any value of z z, we can obtain the corresponding values for , we can obtain the corresponding values for

xx and and yy, , all of which would be solutions to the systemall of which would be solutions to the system.. This system has This system has infinitely many solutionsinfinitely many solutions..


Applications of Systems of Linear EquationsApplications of Systems of Linear Equations


Example: Example: BlendingBlending

Arctic Juice Company makes three juice blends: Arctic Juice Company makes three juice blends: PineOrangePineOrange, using , using 2 quarts2 quarts of of pineapplepineapple juice and juice and 22 quartsquarts of of

orange orange juice per gallon.juice per gallon. PineKiwiPineKiwi, using , using 3 quarts3 quarts of of pineapple pineapple juice and juice and 1 quart1 quart of of

kiwikiwi juice per gallon. juice per gallon. OrangeKiwiOrangeKiwi, using , using 3 quarts3 quarts of of orangeorange juice and juice and 1 quart1 quart of of

kiwikiwi juice per gallon. juice per gallon. Each day the company has Each day the company has 800 quarts800 quarts of of pineapplepineapple juice, juice, 650 650

quartsquarts of of orangeorange juice, and juice, and 350 quarts350 quarts of of kiwikiwi juice available. juice available. How many gallons of each blend should it make each day if it How many gallons of each blend should it make each day if it

wants to wants to use up all suppliesuse up all supplies??



Solution:Solution: First First define the variablesdefine the variables::

xx == number of gallons of PineOrange made each day. number of gallons of PineOrange made each day.yy = = number of gallons of PineKiwi made each day. number of gallons of PineKiwi made each day.zz == number of gallons of OrangeKiwi made each day. number of gallons of OrangeKiwi made each day.


Next, Next, organizeorganize the given information in a the given information in a tabletable::


PineOrange PineOrange ((xx))

PineKiwi PineKiwi ((yy))

OrangeKiwi OrangeKiwi ((zz))

Total Total AvailableAvailable

Pineapple Juice (qt)Pineapple Juice (qt) 22 33 00 800800

Orange Juice (qt)Orange Juice (qt) 22 00 33 650650

Kiwi Juice (qt)Kiwi Juice (qt) 00 11 11 350350

Translate table into a system of equations and solve:Translate table into a system of equations and solve:

2 3 800

2 3 650

350

x y

x z

y z



The solution to the system is:The solution to the system is:

This means Arctic Juice should make This means Arctic Juice should make 100 gallons100 gallons of of PineOrangePineOrange, , 200 gallons200 gallons of of PineKiwiPineKiwi, and , and 150 gallons150 gallons of of OrangeKiwiOrangeKiwi each day. each day.

( , , ) (100,200,150)x y z

For practice, For practice, solve this problem at homesolve this problem at home on your own. See if on your own. See if you get the same solution.you get the same solution.


Example: Example: Aircraft PurchasingAircraft Purchasing

An airline is considering the purchase of aircraft to meet an An airline is considering the purchase of aircraft to meet an estimated demand for 3200 seats.estimated demand for 3200 seats.

The airline has decided to buy:The airline has decided to buy: Boeing 747sBoeing 747s, which seat , which seat 400400 passengers and are priced at passengers and are priced at

$200 million$200 million each. each. Boeing 777sBoeing 777s, which seat , which seat 300300 passengers and are priced at passengers and are priced at

$160 million$160 million;; Airbus A321sAirbus A321s, which seat , which seat 200200 passengers and are priced at passengers and are priced at

$60 million$60 million.. The airline has a budget of The airline has a budget of $1,540$1,540 million for the aircraft and million for the aircraft and

wishes to buy three times as many wishes to buy three times as many 777s 777s as as 747s747s.. How many of each should it order to meet demand?How many of each should it order to meet demand?



Solution:Solution: First First define the variablesdefine the variables::

xx == number of number of BoeingBoeing 747s747s ordered. ordered.yy = = number of number of BoeingBoeing 777s777s ordered. ordered.zz == number of number of AirbusAirbus A321sA321s ordered. ordered.


Next, Next, organizeorganize the given information in a the given information in a tabletable::


This table translates into the following system of equations:This table translates into the following system of equations:

400 300 200 3200

200 160 60 1540

x y z

x y z

Boeing 747 Boeing 747 ((xx))

Boeing 777 Boeing 777 ((yy))

Airbus A321Airbus A321((zz))

Total Total AvailableAvailable

PassengersPassengers 400400 300300 200200 32003200

Cost ($millions)Cost ($millions) 200200 160160 6060 15401540

The problem is that we have The problem is that we have three unknownsthree unknowns, but only , but only two two equationsequations. We need to obtain a . We need to obtain a third equationthird equation..


Remember the airline wishes to purchase Remember the airline wishes to purchase three times as manythree times as many 777s777s as as 747s747s. This gives us with the . This gives us with the third equationthird equation we need: we need:


Thus, we have the following system of Thus, we have the following system of three equations and three equations and three unknownsthree unknowns to solve the problem: to solve the problem:

400 300 200 3200

200 160 60 1540

3 0

x y z

x y z

x y

3 3 0 or: y x x y




This means the airline should order This means the airline should order twotwo 747s747s, , sixsix 777s777s, and , and threethree A321sA321s..

( , , ) (2,6,3)x y z

Again for practice, Again for practice, solve this problem at homesolve this problem at home on your own. on your own. See if you get the same solution.See if you get the same solution.


Example: Example: TransportationTransportation

A car rental company has A car rental company has four locationsfour locations in the city: in the city: NorthsideNorthside, , EastsideEastside, , SouthsideSouthside, and , and WestsideWestside..

WestsideWestside and and EastsideEastside have respectively have respectively 2020 and and 1515 more cars more cars than they need. than they need. NorthsideNorthside and and SouthsideSouthside need respectively need respectively 1010 and and 2525 more cars than they have. more cars than they have.

It costs It costs $10$10 (in salary and gas) to have an employee drive a car (in salary and gas) to have an employee drive a car from from WestsideWestside to to NorthsideNorthside; it costs ; it costs $5$5 to drive a car from to drive a car from EastsideEastside to to NorthsideNorthside; it costs ; it costs $20$20 to drive a car from to drive a car from WestsideWestside to to SouthsideSouthside; and it costs $10 to drive a car from ; and it costs $10 to drive a car from EastsideEastside to to SouthsideSouthside..

If the company will spend a total of $475 rearranging its cars, If the company will spend a total of $475 rearranging its cars, how many cars will it drive from each of how many cars will it drive from each of WestsideWestside and and EastsideEastside to each of to each of NortsideNortside and and SouthsideSouthside??



We have the following unknowns based on the figure:We have the following unknowns based on the figure:

xx == number of cars driven from number of cars driven from Westside Westside toto Northside Northside..yy == number of cars driven from number of cars driven from Eastside Eastside toto Northside Northside..zz == number of cars driven from number of cars driven from Westside Westside toto Southside Southside..ww == number of cars driven from number of cars driven from Eastside Eastside toto Southside Southside

Northside

Southside

Westside Eastside

x

z

y

w

$10/car

$20/car

$5/car

$10/car


Consider the Consider the NorthsideNorthside location. It needs location. It needs 1010 more cars, so the more cars, so the total number of cars being driven to total number of cars being driven to NorthsideNorthside should be should be 1010::


Similarly, the total number of cars driven to Similarly, the total number of cars driven to SouthsideSouthside should should be be 2525::

Considering the number of cars that can be driven out of Considering the number of cars that can be driven out of WestsideWestside and and EastsideEastside we get: we get:

10x y

25z w

20

15

x z

y w

Finally, the firm will spend Finally, the firm will spend $475$475 for transportation: for transportation:

10 5 20 10 475x y z w


Thus, we have the following system of Thus, we have the following system of five equations and four five equations and four unknownsunknowns to solve the problem: to solve the problem:


10

25

20

15

10 5 20 10 475

x y

z w

x z

y w

x y z w




This means the the company will drive This means the the company will drive 55 cars from cars from WestsideWestside to to NorthsideNorthside, , 55 from from EastsideEastside to to NorthsideNorthside, , 1515 from from WestsideWestside to to SouthsideSouthside, and , and 1010 from from EastsideEastside to to SouthsideSouthside..

( , , , ) (5,5,15,10)x y z w

Again for practice, Again for practice, solve this problem at homesolve this problem at home on your own. on your own. See if you get the same solution.See if you get the same solution.


Dude, that was like 76slides. I thought this would be EZ!


A solution to boredom?A solution to boredom?

You can find the key to EZ in the definition from the 2You can find the key to EZ in the definition from the 2ndnd slide: slide: Matrices can be used to Matrices can be used to representrepresent systems of equationssystems of equations and to and to

solve themsolve them systematicallysystematically.. Note the word in yellow. When you see that word, you can use Note the word in yellow. When you see that word, you can use

a computer system to do all the work. a computer system to do all the work. Computers are really good at doingComputers are really good at doing

things systematically, because they arethings systematically, because they arereally good at boring, repetitious really good at boring, repetitious tasks!tasks!

W&C - Chapter 2

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