UNIT 1. RATIOS & PROPORTIONS
UNIT 1. RATIOS &PROPORTIONS
RATIOSRatios are comparisons made between two setsof numbers.For example:There are eight girls and seven boys in a class.The ratio of girls to boys is 8 to 7.
Ratios are used everyday. They are used for: Miles per hour The cost of items per pound, gallon, etc. Hourly rate of pay80 miles to 1 hour 80mph
THERE ARE 3 WAYS TO WRITE RATIOS.1. Write the ratio using the word “to” between the twonumbers being compared.For example: There are 8 girls and 5 boys in my class.What is the ratio of girls to boys?The ratio is: 8 girls to 5 boys8 to 5
2. Write a ratio using a colon between the twonumbers being compared.For example: There are 3 apples and 4 oranges in thebasket. What is the ratio of apples to oranges?The ratio is: 3 apples to 4 oranges.3:4
3. Write a ratio as a fraction.For example:Hunter and Brandon were playing basketball. Brandonscored 5 baskets and Hunter scored 6 baskets. Whatwas the ratio of baskets Hunter scored to the basketsBrandon scored?The ratio of baskets scored was:6 baskets to 5 baskets65
GUIDED PRACTICE:Directions: Write the ratio in three different ways.There are 13 boys and 17 girls in sixth grade.Find the ratio of boys to the girls in sixth grade.13 to 1713 : 171317
RULES FOR SOLVING RATIO PROBLEMS.1.When writing ratios, the numbers should be written inthe order in which the problem asks for them.For example: There were 4 girls and 7 boys at the birthdayparty.What is the ratio of girls to boys?Hint: The question asks for girls to boys; therefore, girlswill be listed first in the ratio.4 girls to 7 boys4 girls : 7 boys4 girls7 boys
GUIDED PRACTICE:Directions: Solve and write ratios in all three forms. 1. The Panthers played 15 games thisseason.They won 13games.What is the ratio of games won to gamesplayed?The questions asks forto Games played.13 to 1513:15Games won 1513
2. Amanda’s basketball team won 7 games and lost 5.What is the ratio of games lost to games won?THE QUESTION ASKS FOR GAMES LOST TO GAMESWON. THEREFORE, THE NUMBER OF GAMES LOSTSHOULD BE WRITTEN FIRST, AND THE GAMES WONSHOULD BE WRITTEN SECOND.Games lost 5 to Games won 75 to 75:757
REDUCING RATIOSRatios can be reduced withoutchanging their relationship.2 boys to 4 girls 1 boy to 2 girls
REDUCING RATIOSIs this relationshipthe same?2 boys to 4 girls 1 boy to 3 girls
2. ALL RATIOS MUST BE WRITTEN INLOWEST TERMS.Steps:1. Read the word problem.2. Set up the ratio.For example:You scored 40 answers correct out of 45 problems on atest. Write the ratio of correct answers to total questions inlowest form.Step 1: Read the problem. What does it want to know?40 to 4540 : 454045
3. Reduce the ratio if necessary.Reduce means to break down a fraction or ratio into thelowest form possible.Reduce smaller number; operation will always be division.HINT: When having to reduce ratios, it is better to set up the ratio in thevertical form. (Fraction Form)4045Look at the numbers in the ratio. What ONEnumber can you divide BOTH numbers by?40 to 45 Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40Factors of 45: 1, 3, 5, 9, 15, 4540 45 55 89
1. There are 26 black cards in a deck of playing cards. If there are 52 cardsin a deck, what is the ratio of black cards to the deck of cards?Step 1: Read the problem. (What does it want to know?)Step 2: Set up the ratio.26 black cards to 52 cardsStep 3: Can the ratio be reduced? If so, set it up like a fraction.26 52 2626 12What is the largest number that will go intoboth the top number and the bottom numberevenly? (It can not be the number one!)
PROPORTIONSProportions are two ratios of equal value.1 girl4 boys4 girls16 boysAre these ratios saying the same thing?
PROPORTIONSProportions are two ratios of equal value.1 girl4 boys5 girls16 boysAre these ratios proportions?
DETERMINING TRUE PROPORTIONS:To determine a proportion true, cross multiply.If the cross products are equal, then it is a true proportion.4520 x 5100 2025 4 x 25 100The cross products were equal, therefore 4 And 20 makes a true proportion.525
Guided Practice:Directions: Solve to see if each problem is a true proportion.1. 3 155252.3.6 578767 3712 60
Guided Practice:Directions: Solve to see if each problem is a true proportion.315 1. 52515 x 5 3 x 2575 true752.68 577657 x 8 6 x 76456 true4563. 712 37607 x 60 37 x 12420 false444
SOLVING PROPORTIONS WITH VARIABLESWhat is a variable?A variable is any letter that takes place of amissing number or information.Eric rode his bicycle a total of 52 miles in 4 hours. Riding atthis same rate, how far can he travel in 7 hours?Look for the two sets ofratios to make up aproportion.Set 1The proportion should beset equal to each other.You have 52 miles in 4hours. This is the firstratio.52 miles4 hoursNext, the problem states “howfar can he travel in 7 hours.The problem is missing themiles. Therefore, the milesbecomes the variable.Set 2n miles7 hours52 n74HINT: The order of the ratio does matter!
SOLVING THE PROPORTION:When solving proportions, follow these rules:1. Cross multiply.2.Divide BOTH sides by the number connected to the variable. 3.Check the answer to see if it makes a true proportion.Problem:52 44xnWhich number isconnected to the variable?Since the 4 is connectedto the variable, DIVIDEboth sides by the 4. 52 x 74n 4n7364n 91 miles44 4 1;364 4 therefore youare left with “n”on one side.91
If it comes out even, then the answer is correct.Check your answer!524 91752 x 7 91 x 4364 364
GUIDED PRACTICEDirections: Solve each proportion.1.For every dollar Julia spends on her MasterCard, she earns 3 frequent flyer miles withAmerican Airlines. If Julia spends 609 dollars onher card, how many frequent flyer miles will sheearn?
GUIDED PRACTICEDirections: Solve each proportion.1. For every dollar Julia spends on her Master Card, sheearns 3 frequent flyer miles with American Airlines. If Juliaspends 609 dollars on her card, how many frequent flyermiles will she earn?Step 1: Set up the proportion.Step 2: Cross multiply.Step 3: Divide 1.00 3 miles1d1Step 4: Check answer.d 609.00d miles 18271 1827
1. Justin’s car uses 40 gallons of gas to drive 250miles. At this rate, approximately, how manygallons of gas will he need for a trip of 600 miles.2. If 3 gallons of milk cost 9, how many jugs canyou buy for 45?3.On Thursday, Karen drove 400 miles in 8 hours.At this same speed, how far can she drive in 12hours?
1. Justin’s car uses 40 gallons of gas to drive 250miles. At this rate, approximately, how manygallons of gas will he need for a trip of 600 miles.40 galx gal250 mi 600mi40250 x600250x 24000250x250x 24000250 96Check:4025024000 9660024000
2. If a 3 gallon jug of milk cost 9, how many 3gallon jugs can be purchased for 45?1 n9 451 n9 451x45 9n459 9n95 n5 jugs of milk can bepurchased for 45Check:1 945545 45
3.On Thursday, Karen drove 400 miles in 8 hours. At thissame speed, how far can she drive in 12 hours?400 miles8 hours x124008 x128x 48004008x x miles12 hours600 miles
4.Susie has two flower beds in which to plant tulips anddaffodils. She wants the proportion of tulips to daffodils to bethe same in each bed. Susie plants 10 tulips and 6 daffodils inthe first bed. How many tulips will she need for the second bedif she plants 15 daffodils?10 tulips 6 daffodils106 x156x 150 15066x6x 2510 x156x tulips15 daffodilsx 25 tulips10 6150 2515150
Contents - Percentages What are Percentages ? Converting between Fractions and Percentages Finding a Percentage
What are Percentages ? They are just like fractions The whole is always split into 100 parts So 35% means 35 out of the 100 parts35100
Finding a Percentage Use this info to find any percentage i.e. Find 30% of 240 10% 24 So, 30% 3 x 24 72 30% 72.00Without aculator
Unit 2: Profit & Loss
How business works Shops buy goods from suppliers at cost price.COST PRICE: 40
Cummins Sports then sell the good to the consumer at a higher price. Thisis known as the selling price.THE DIFFERENCE BETWEEN THE COST PRICE ANDTHE SELLING PRICE IS CALLED PROFIT
Profit & Profit Percentage Question: How much profit did Cummins Sports makeon the jersey? Answer: 20 Companies express the profit they make inpercentage figures rather than in money. This is known as a percentage profit or it can also bea percentage loss.Question: What type of situation would a shop have apercentage loss?
Percentage Profit & LossPERCENTAGE PROFIT (OR LOSS)
Percentage Profit Q: What percentage profit did Cummins Sports make onselling the Cork jersey? Cost Price: 40 Selling Price: 60 Profit: 20 Percentage Profit Percentage Profit
Questions A shop buys Justin Bieberfigures in boxes of 50 units.The shop purchases 10 boxes ofdolls.The total cost of purchasingthe 10 boxes was 6000.Each doll was sold at a price of 15Q: Find the following:(i) Cost Price of each doll(ii) The profit made on eachdoll.(iii) The profit percentage madeon each doll.
Answers(i)Cost Price 6000500dollsCost Price 12 per doll(ii)Profit 15 - 12 3 profit on each doll(iii) Profit Percentage 3 100x 25% 12 1Pr ofit100xCost Price 1
Questions The Carphone Warehouse bought 50iPhone 4S @ 400 each.They sold 40 of them at the markedprice of 600.The remaining were sold during asale at a discount of 10%.Questions:(i) How much was paid for the 50iPhones?(ii)Find the selling price of each ofthe iPhones during the sale.(iii)Find the total profit on selling the50 iPhones.(iv)Find the percentage profit made.
REMEMBERCOST PRICE 100%ALWAYS WORK FROM THERE
REMEMBERPercentage Profit (or loss) Pr ofit100xCost Price 1
Newcastle Football PlayerDemba Ba wears Adidas F50adiZero Football Boots.The selling price of thesefootball boots are 180.This selling price includes aprofit of 20%.Questions:(i)Find the cost price of thefootball boots.(ii)Find the percentage profitmade if the boots were soldat 165.
Answers(i)120% 1801% 180/1201% 1.50100% (Cost Price) 150(ii)Boots Sold @ 65Percentage Profit Percentage Profit Percentage Profit 10%Pr ofit100xCost Pr ice1 15100x 1501
Question Pennys are having a huge sale toclear some old stock from lastyear.A pair of shoes are being sold at 3.60When selling at this price Pennysmake a loss of 10%.Questions:(i)Find the cost price of theshoes.(ii)Find the percentage loss ifPennys discount the shoes by afurther 10%.
Answers(i)Selling Price 3.60 (10% Loss)90% 3.601% 0.04 (4 cents)100% (Cost Price) 4.00(ii) 3.60 – 10% 3.60 - 0.36 3.24 (New Selling Price)Total Loss 4.00 - 3.24 0.76Percentage Loss Loss100xCost Pr ice1Percentage Loss 0.76 19% 4.00
UNIT 3:Simple and CompoundInterest
Warm Up: Find 6% of 400. 24 Find 5% of 2,000. 100 Find 4.5% of 700. 31.50 Find 5.5% of 325. 17.88
Simple Interest When you first deposit money in a savingsaccount, your deposit is calledPRINCIPAL. The bank takes the money and invests it. In return, the bank pays you INTERESTbased on the INTEREST RATE. Simple interest is interest paid only on thePRINCIPAL.
Simple Interest FormulaI prt I interestP principalR the interest rate per yearT the time in years.
Real-World Suppose you deposit 400 in a savings account.The interest rate is 5% per year. Find the interest earned in 6 years. Find the totalof principal plus interest. I PRT Formula P 400, R 0.05 5%, T 6 (in years) 400 x 0.05 20 interest on one year 400 x 0.05 x 6 120 interest on 400 over 6years 400 120 520 amount in account after 6years.
Now Figure Interest In Months Remember that T time in Years. So, Find the interest earned in three months. Find thetotal of principal plus interest. What fraction of a year is 3 months?T 3/12 ¼ or 0.25I PRTI 400 x 0.05 x 0.25I 5 interest earned after 3 months 5 400 total amount in account 405
Try These: Both Findthe Simple Interest Principal 250 Interest Rate 4% Time 3 Years 30 Principal 250 Interest Rate 3.5% Time 6 Months 4.38Reminder: Time is alwaysin terms of Years. So, ifyou’re dealing withmonths, you have to makeyour months a fraction of ayear.
Compound Interest Compound Interest is when the bank paysinterest on the Principal AND the Interestalready earned. The Balance is the Principal PLUS theInterest. The Balance becomes the Principal onwhich the bank figures the next interestpayment when doing Compound Interest.
Compound Interest Example You deposit 400 in an account that earns5% interest compounded annually (once peryear). What is the balance in your accountafter 4 years? In your last calculation, roundto the nearest cent.
Fill In This ChartPrinciple @Beginning ofYearYear 1: 400.00Year 2:Year 3:Year 4:Interest(I PRT)Balance at Endof Each Year 486.20
Compound Interest Formula You can find a balance using compoundinterest in one step with the compoundinterest formula. An INTEREST PERIOD is the length oftime over which interest is calculated. The Interest Period can be a year or lessthan a year.
Compound Interest FormulaB p(1 r)nB the final balanceP is the principalR the interest rate for each interest periodN the number of interest periods.
Semi-Annual When interested is compoundedsemiannually (twice per year), you mustDIVIDE the interest rate by the number ofinterest periods, which is 2.6% annual interest rate 2 interest periods 3% semiannual interest rateTo find the number of payment periods,multiply the number of years by the numberof interest periods per year.
Example Find the balance on a deposit of 1,000,earning 6% interest compoundedsemiannually for 5 years. The interest rate R for compoundingsemiannually is 0.06 2, or 0.03. Thenumber of payment periods N is 5 years x 2interest periods per year, or 10. Now plug it into the formula!
The Formula!B p (1 R)nB 1,000 (1 0.03)10B 1,000 (1.03)10B 1,000 (1.34391638)B 1,343.92Happy? You’ll actually get to use acalculator for these ]
Fractions & Percentages
Contents Fractions – the Language of Equivalent Fractions and Cancelling Fractions Top Heavy Fractions and Mixed Numbers Ordering Fractions Finding a Fraction
What are Fractions ? Parts of a whole Eg.123423Numerator2 Denominator3Unit Fractions1 1 1 1 12 3 4 5 5067
Fractions 12What do they mean We have 1 of those partsThe whole is split into 2 partsWe have 3 of those parts34The whole is split into 4 parts
Equivalent Fractions1224341216x212 x210/11/2020x42434 35x41216
Equivalent Fractionsx458x3203x4712 x32136
Cancelling Fractions Reducingfractions to their simplestform5 Equivalent fractions using the10smallest numbers Reduceto simplest form Find H C5F of 5 and 10 So 5divide b1oth numerator &denominator by 510 525
Cancelling Fractions Cancel HCF 3 So,16361215HCF 4 31215 3 4451636 449
Top Heavy Fractions & Mixed Numbers2173 Convert 1 125into a mixed number1 12 5 2 with left over2525210/11/20202572
Top Heavy Fractions & Mixed Numbers 26into3a mixed numberConvert2left over, so3 26 3 8, with Convert How many fifths8into a top heavy fraction33518510/11/20207323
Top Heavy Fractions & Mixed Numbers3 335335Convert So Tryinto a top heavy fraction(3x5) 3 513 (3x3) 103185 537 (5x7) 387
Ordering Fractions1423Easy !! But what about . Con vert to equiv al entfractions with same denominator first 35 11Find LCM of 5, 8 and 20 (denominators)240
Ordering Fractions 32558Now put them in order So, then answer the question2241120 211202442543558 2240
Finding a Fraction1 What is What about To find 5 To find 25 2 50of 50 ?14 100of 100 ?11How would we find21157yes, 157
Finding a Fraction2 What aboutof 125 ?1 Findfirst – Then, x by 2 So,of 125 is 502255125x250
UNIT 4. Matrices andDeterminant
Matrices - IntroductionMatrix algebra has at least two advantages: Reduces complicated systems of equations to simple expressions Adaptable to systematic method of mathematical treatment and wellsuited to computersDefinition:A matrix is a set or group of numbers arranged in a square or rectangulararray enclosed by two brackets 1 1 4 2 3 0 a b c d
Matrices - IntroductionProperties: A specified number of rows and a specified number of columns Two numbers (rows x columns) describe the dimensions or size of thematrix.Examples:3x3 matrix2x4 matrix1x2 matrix 1 2 4 4 1 5 1 1 3 3 3 0 03 3 23 1 1
Matrices - IntroductionA matrix is denoted by a bold capital letter and the elements within thematrix are denoted by lower case letterse.g. matrix [A] with elements aijAmxn mAn a11 a 21 ⁝ am1i goes from 1 to mj goes from 1 to na12. aija22. aij⁝⁝am2aijain a2n ⁝ amn
Matrices - IntroductionTYPES OF MATRICES1. Column matrix or vector:The number of rows may be any integer but the number ofcolumns is always 1 1 4 2 1 3 a11 a21 ⁝ am1
Matrices - IntroductionTYPES OF MATRICES2. Row matrix or vectorAny number of columns but only one row 1 01 a11a12a133a1n
Matrices - IntroductionTYPES OF MATRICES3. Rectangular matrixContains more than one element and number of rows is not equal to thenumber of columns 1 1 3 7 7 7 7 6 1 1 1 0 0 2 0 3 3 0 m n
Matrices - IntroductionTYPES OF MATRICES4. Square matrixThe number of rows is equal to the number of columns(a square matrix A has an order of m)mxm 1 1 3 0 1 1 1 9 9 0 6 6 1 The principal or main diagonal of a square matrix is composed of all elements aijfor which i j
Matrices - IntroductionTYPES OF MATRICES5. Diagonal matrixA square matrix where all the elements are zero except those on the maindiagonal 1 0 0 0 2 0 0 0 1 i.e. aij 0 for all i jaij 0 for some or all i j 3 0 0 0 3 0 0 0 5 0 0 00 0 0 9
Matrices - IntroductionTYPES OF MATRICES6. Unit or Identity matrix - IA diagonal matrix with ones on the main diagonal 1 0 0 0 1 0 0 0 1 0 0 0i.e. aij 0 for all i jaij 1 for some or all i j0 0 0 1 1 0 0 1 aij 0 0 aij
Matrices - IntroductionTYPES OF MATRICES7. Null (zero) matrix - 0All elements in the matrix are zero 0 0 0 aij 0 0 0 0 0 0 0 0 0 0 For all i,j
Matrices - IntroductionTYPES OF MATRICES8. Triangular matrixA square matrix whose elements above or below the main diagonal are allzero 1 0 0 2 1 0 5 2 3 1 0 0 2 1 0 5 2 3 1 8 9 0 1 6 0 0 3
Matrices - IntroductionTYPES OF MATRICES8a. Upper triangular matrixA square matrix whose elements below the main diagonal are allzero aij 0 0 aijaai.e. aij 0 for all i j 1 8 7 0 1 8 0 0 3 1 7 4 0 1 7 0 0 7 0 0 04 4 8 3
Matrices - IntroductionTYPES OF MATRICES8b. Lower triangular matrixA square matrix whose elements above the main diagonal are all zero aij a ij aij0aijaiji.e. aij 0 for all i j0 0 aij 1 0 0 2 1 0 5 2 3
Matrices – IntroductionTYPES OF MATRICES9. Scalar matrixA diagonal matrix whose main diagonal elements are equal to thesame scalarA scalar is defined as a single number or constant aij 0 000ai.e. aij 0 for all i jaij a for all i j 1 0 0 0 1 0 0 0 1 6 0 0 0 6 0 0 0 6 0 0 00 0 0 6
MatricesMatrix Operations
Matrices - OperationsEQUALITY OF MATRICESTwo matrices are said to be equal only when all correspondingelements are equalTherefore their size or dimensions are equal as wellA 1 0 0 2 1 0 5 2 3 B 1 0 0 2 1 0 5 2 3 A B
Matrices - OperationsSome properties of equality: IIf A B, then B A for all A and B IIf A B, and B C, then A C for all A, B and CA If A B then 1 0 0 2 1 0 5 2 3 aij bijB b11 b12 b13 b bb 21 22 23 b31 b32 b33
Matrices - OperationsADDITION AND SUBTRACTION OF MATRICESThe sum or difference of two matrices, A and B of the same size yields amatrix C of the same sizecij aij bijMatrices of different sizes cannot be added or subtracted
Matrices - OperationsCommutative Law:A B B AAssociative Law:A (B C) (A B) C A B C5 6 88 5 7 3 1 1 2 5 6 4 2 3 2 7 9 A2x3B2x3C2x3
Matrices - OperationsA 0 0 A AA (-A) 0 (where –A is the matrix composed of –aij as elements) 6 4 2 1 2 0 5 2 2 3 2 7 1 0 8 2 2 1
Matrices - OperationsSCALAR MULTIPLICATION OF MATRICESMatrices can be multiplied by a scalar (constant or single element)Let k be a scalar quantity; thenkA AkEx. If k 4 and 3 2A 2 4 1 1 3 1
Matrices - OperationsProperties: k (A B) kA kB (k g)A kA gA k(AB) (kA)B A(k)B k(gA) (kg)A
Matrices - OperationsMULTIPLICATION OF MATRICESThe product of two matrices is another matrixTwo matrices A and B must be conformable for multiplication to be possiblei.e. the number of columns of A must equal the number of rows of BExample.A(1x3)xB (3x1)C(1x1)
Matrices - OperationsB xA Not possible!(2x1) (4x2)Ax(6x2)B Not possible!(6x3)ExampleA(2x3)xB(3x2) C(2x2)
Matrices - Operations a11 a12 a 21 a22 b11a13 b a b12 (a11 b11 ) (a12 b21 ) (a13 b31 ) c11(a11 b12 ) (a12 b22 ) (a13 b32 ) c12(a21 b11 ) (a22 b21 ) (a23 b31 ) c21(a21 b12 ) (a22 b22 ) (a23 b32 ) c22Successive multiplication of row i of A with column j of B – row bycolumn multiplication
Matrices - Operations 4 8 1 2 3 (1 4) (2 6) (3 5) (1 8) (2 2) (3 3) 2 7 6 2 4(4 4) (2 6) (7 5)(4 8) (2 2) (7 3) 5 3 31 2 Remember also:IA A 1 0 31 21 0 1 63 57 31 21 63 57
Matrices - OperationsAssuming that matrices A, B and C are conformable forthe operations indicated, the following are true:1. AI IA A2. A(BC) (AB)C ABC -(associative law)3. A(B C) AB AC - (first distributive law)4. (A B)C AC BC - (second distributive law)Caution!1. AB not generally equal to BA, BA may not be conformable2. If AB 0, neither A nor B necessarily 03. If AB AC, B not necessarily C
Matrices - OperationsAB not generally equal to BA, BA may not be conformable 1 2 T 50 3 4 S 02 1 2 3TS 050 3 4 1ST 502 4 3 8 2 15 20 2 23 6 0 10 0
Matrices - OperationsIf AB 0, neither A nor B necessarily 03 0 0 1 1 2 0 0 2 3 0 0
Matrices - OperationsTRANSPOSE OF A MATRIXIf : 2 4 7 A 2 A 2x351 3 3Then transpose of A, denoted AT is:A 2 AT3T 2 5 4 3 7 1 a aijTjiFor all i and j
Matrices - OperationsTo transpose:Interchange rows and columnsThe dimensions of AT are the reverse of the dimensions of A 2 4 7 A 2 A 51 3 3T2A 3 AT 2 5 4 3 7 1 2x33x2
Matrices - OperationsProperties of transposed matrices:1. (A B)T AT BT2. (AB)T BT AT3. (kA)T kAT4. (AT)T A
Matrices - Operations1. (A B)T AT BT 7 3 1 1 2 5 6 56 82 1 4 8 2 7 3 5 5 2 8 7 1 6 6 3 5 9 8 2 8 5
Matrices - Operations(AB)T BT AT 1 1 1 0 2 0 2 3 1 2 8 8 2 1 0 1 1 2 1 2 2 8 0 3
Matrices - OperationsSYMMETRIC MATRICESA Square matrix is symmetric if it is equal to its transpose:A AT aA bb d a b A bd T
Matrices - OperationsWhen the original matrix is square, transposition does not affect theelements of the main diagonal aA cb d a c A bd TThe identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal totheir transpose since the diagonal is unaffected.
Matrices - OperationsINVERSE OF A MATRIXConsider a scalar k. The inverse is the reciprocal or division of 1 by the scalar.Example:k 7 the inverse of k or k-1 1/k 1/7Division of matrices is not defined since there may be AB AC while B CInstead matrix inversion is used.The inverse of a square matrix, A, if it exists, is the unique matrix A-1 where:AA-1 A-1 A I
Matrices - OperationsExample: 3A 2 A 2 1 1A 22Because:1 1 1 3 1 1 3 1 1 2 3 2 1 0 3 1 1 1 1 2 1 2 3 0 0 1 0 1
Matrices - OperationsProperties of the inverse:( AB) 1 B 1 A 1( A 1 ) 1 A( AT ) 1 ( A 1 )T1 1 1(kA) AkA square matrix that has an inverse is called a nonsingular matrixA matrix that does not have an inverse is called a singular matrixSquare matrices have inverses except when the determinant is zeroWhen the determinant of a matrix is zero the matrix is singular
Matrices - OperationsDETERMINANT OF A MATRIXTo compute the inverse of a matrix, the determinant is requiredEach square matrix A has a unit scalar value called the determinant of A,denoted by det A or A Ifthen 1 2 A 65 1 2A 6 5
Matrices - OperationsIf A [A] is a single element (1x1), then the determinant is defined as thevalue of the elementThen A det A a11If A is (n x n), its determinant may be defined in terms of order (n-1) or less.
Matrices - OperationsMINORSIf A is an n x n matrix and one row and one column are deleted, theresulting matrix is an (n-1) x (n-1) submatrix of A.The determinant of such a submatrix is called a minor of A and isdesignated by mij , where i and j correspond to the deletedrow and column, respectively.mij is the minor of the element aij in A.
Matrices - Operationseg. a11 a12 A a 21 a22 a31 a32a13 a23 a33 Each element in A has a minorDelete first row and column from A .The determinant of the remaining 2 x 2 submatrix is the minor of a11a22m11 a32a23a33
Matrices - OperationsTherefore the minor of a12 is:a21 a23m12 a31 a33And the minor for a13 is:a21 a22m13 a31 a32
Matrices - OperationsCOFACTORSThe cofactor Cij of an element aij is defined as:C ( 1) i j mijijWhen the sum of a row number i and column j is even, cij mij and when i j isodd, cij -mijc (i 1, j 1) ( 1)1 1 m m111111c (i 1, j 2) ( 1)1 2 m m1212121313c (i 1, j 3) ( 1) m m1 313
Matrices - OperationsDETERMINANTS CONTINUEDThe determinant of an n x n matrix A can now be defined asA det A a11c11 a12c12 a1n c1nThe determinant of A is therefore the sum of the products of the elements ofthe first row of A and their corresponding cofactors.(It is possible to define A in terms of any other row or column but forsimplicity, the first row only is used)
Matrices - OperationsTherefore the 2 x 2 matrix : a11 a12 A a 21 a22 Has cofactors :c11 m11 a22 a22And:c12 m12 a21 a21And the determinant of A is:A a11c11 a12c12 a11a22 a12 a21
Matrices - OperationsExample 1: 3 1 A 12 A (3)(2)
Matrices - OperationsFor a 3 x 3 matrix: a11 a12A a 21 a22 a31 a32a13 a23 a33 The cofactors of the first row are:a22c11 a32a21c12 a31a21c13 a31a23 a22 a33 a23a32a33a23 (a21a33 a23a31 )a33a22 a21a32 a22 a31a32
Matrices - OperationsThe determinant of a matrix A is:A a11c11 a12c12 a11a22 a12 a21Which by substituting for the cofactors in this case is:A a11(a22a33 a23a32 ) a12 (a21a33 a23a31) a13(a21a32 a22a31)
Matrices - OperationsExample 2: 1 0 1 A 0 2 3 1 0 1 A a11(a22a33 a23a32 ) a12 (a21a33 a23a31) a13(a21a32 a22a31)A (1)(2 0) (0)(0
Matrices - OperationsADJOINT MATRICESA cofactor matrix C of a matrix A is the square matrix of the same order as A inwhich each element aij is replaced by its cofactor cij .Example:If 1 2 A 34 The cofactor C of A is 4 3 C 21
Matrices - OperationsThe adjoint matrix of A, denoted by adj A, is the transpose of its cofactormatrixadjA CTIt can be shown that:A(adj A) (adjA) A A IExample: 1 2 A 34 A (1)(4) (2)( 3) 10 4 2 adjA C 31 T
Matrices - Operations 1 2 4 2 10 0 A(adjA) 10I 3 4 3 1 0 10 4 2 1 2 10 0 (adjA) A 10I 3 1 3 4 0 10
Matrices - OperationsUSING THE ADJOINT MATRIX IN MATRIX INVERSIONSinceAA-1 A-1 A IandA(adj A) (adjA) A A IthenA 1 adjAA
Matrices - OperationsExampleA 1 2 3 4 1 4 2 A 10 1To check 0.4AA-1 A-1 A I0.4 0.2 1 1 2 AA 3 4 0.3 0.1 0 0.4 0.2 1 2 1 1A A 0.3 0.1 3 4 0 10 I 1 0 I 1
Matrices - OperationsExample 2 3 1 1 A 2 1 0 1 2 1 The determinant of A is A (3)(-1-0)-(-1)(-2-0) (1)(4-1) -2The elements of the cofactor matrix arec11 ( 1),c12 ( 2),c13 (3),c21 ( 1),c22 ( 4),c23 (7),c31 ( 1),c32 ( 2),c33 (5),
Matrices - OperationsThe cofactor matrix is therefore3 1 2C 1 4 7 1 25 soand 1 1 1 adjA C T 2 4 2 3 7 5 1 1 1 0.5 0.5 0.5 adjA 1 1 2.0 1.0A 2 4 2 1.0 A 2 3 7 5 1.5 3.5 2.5
Matrices - OperationsThe result can be checked usingAA-1 A-1 A IThe determinant of a matrix must not be zero for the inverse to exist as therewill not be a solutionNonsingular matrices have non-zero determinantsSingular matrices have zero determinants
Matrix InversionSimple 2 x 2 case
Simple 2 x 2 caseLet aA band wA y 1Since it is known thatA A-1 Ithen a b w c d y x 1 0 z 0 1 x z
Simple 2 x 2 caseMultiplying givesaw by 1ax bz 0cw dy 0cx dz 1It can simply be shown thatA ad bc
Simple 2 x 2 casethus1 awy b cwy d1 aw cw bddd w da bc A
Simple 2 x 2 case axz b1 cxz d ax 1 cx bdbb x A da bc
Simple 2 x 2 case1 byw a dyw c1 by dy accc y ad cbA
Simple 2 x 2 case bzx a1 dzx c bz 1 dz acaa z ad bc A
Simple 2 x 2 caseSo that for a 2 x 2 matrix the inverse can be constructed in asimple fashion as w x A y z 1 d A c Ab A 1 d b a A
PROPORTIONS Proportions are two ratios of equal value. 1 girl 4 boys 5 girls 16 boys Are these ratios proportions? DETERMINING TRUE PROPORTIONS: To determine a proportion true, cross multiply. If the cross products are equal, then it is
6th Grade Proportions and Ratios 60 minute connection 2 sites per connection Program Description Math Mania Proportions and Ratios reviews the mathematical concepts of multiplication, division, ratios, percents, and proportions. Students will compete with other classes to solve math probl
6. A summary of the five main categories of selected financial ratios over the period being analyzed are: a. Internal liquidity ratios b. Operating efficiency ratios c. Operating profitability ratios d. Business risk (operating) analysis ratios e. Financial risk (leverage) analysis ratios 7.
Financial ratios can be classified into ratios that measure: (1) profitability, (2) liquidity, (3) management efficiency, (4) leverage, and (5) valuation and growth (Syamsuddin, 2009). In this study, for the purpose of financial ratio analysis, we use four ratios, namely liquidity ratios, activity ratios, leverage ratios, profitability ratios.
activity set can be found. Standard Set title Page number 7.RP.1. Ratios and Proportions 1 7.RP.2. Ratios and Proportions 1 7.RP.2. Analyzing and Describing Relationships 21 7.RP.2. Graphing Relationships 28 7.RP.2. Representing Patterns and Relationships 37 7.RP.3. Ratios and Proportions 1 7.RP.3. Problem Solving with Rational Numbers 14 7.EE.3.
Lesson 7-1 Ratios and Proportions 367 A is a statement that two ratios are equal.You can write a proportion in these forms: and a b c d When three or more ratios are equal, you can write an For example, you could write the following: Two equations are equivalent when either can be
Page 3 UNIT 6 RATIOS RATES PROPORTIONS Meas. CONVERSIONS CCM6 7 Page 3 Today’s Main Ideas: Ratios and Rates Big Idea Definition Example Ratio Rate Unit Rate Dimensional Analysis converting between units .set it up so the right things cancel! Convert 10 mi/h to feet per min
Using Cross Products Video 1, Video 2 Determining Whether Two Quantities are Proportional Video 1, Video 2 Modeling Real Life Video 1, Video 2 5.4 Writing and Solving Proportions Solving Proportions Using Mental Math Video 1, Video 2 Solving Proportions Using Cross Products Video 1, Video 2 Writing and Solving a Proportion Video 1, Video 2
of domestic violence in 2003. Tjaden and Thoennes (2000) found in the National Violence Against Women Survey that 25.5% of women and 7.9% of men self-reported having experienced domestic violence at some point in their lives. Unfortunately, only a small percentage of abused men are willing to speak out in fear of ridicule, social isolation, and humiliation (Barber, 2008). Therefore, because of .
