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“IN THE NAME OF ALLAH THE MOST BENEFICIENT, THE MOST MERCIFUL”

WEL COME TO BUSINESS

MATHEMATICS COURSE

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• In the course the main concentration is on the applications of mathematics to businesses.

• These may be seen in the form of stated problems from the text book.

• Our focus will be on the procedures to find the solutions of these problems.

• Some business terminologies will be used in the course.

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Mathematics of Finance 1Profit:

The difference between the selling price and the cost price of an item is called the profit or mark-up. Thus if “S” is the selling price and “C” is the cost price the mark-up can be calculated as:

P = S – C The mark-up is expressed as the

percentage of the cost price or the selling price. Often the mark-up is based on the cost. The selling price can be determined by

adding the mark-up to the cost.

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Continue…If “C” and “S” represents the cost price and the sale price respectively and “r” is the percentage mark-up, then:

S = C + Cr = C(1+r)Example:

If the Cost price of an item is Rs. 2,400 and the mark-up on cost is 23%. Find the sale price.

Sol:Using the formula

S = C(1+r)Putting the values

S = 2,400(1+0.23) = 2,9525

Continue…

When mark-up is stated as a percentage on sale If “p” is the percentage mark-up on sales. Then p.S would

represent the profit or mark-up.Now

Profit = Sales price – Cost price p.S = S – C

Or C = S - p.S = S(1- p)

Example:After a mark-up of 30% on sales a watch sells for

Rs. 225.i) What is its cost price?ii)What is the %age mark-up on sales if the cost price of

watch would have been Rs. 153? (Solution on White Board) 6

Simple Interest and Present Value

o Interest is a fee which is paid for having the use of money.

oThe amount of money that is lent or invested is called principal.

o Interest is usually paid in proportion to the principal and the period of time over which the money is used.

oThe interest rate specifies the rate at which interest accumulates.

oThe interest rate is typically stated as a percentage of the principal per period of time.

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Continue…

o The interest paid only on the principal is called simple interest.

o Simple interest is usually associated with loans or investments which are short-term in nature.

ComputationSimple interest = (principal ) * ( interest rate) * (number of time period)

Or I = PrtWhere I = simple interest

P = principalr = interest ratet = number of time period of loan

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Continue…

The total amount “A” to be repaid is the principal plus the accumulated interest, or

A = P + I = P + Prt = P(1 + rt)

If the future value “A” is known, the present value of an amount “P” at simple interest “r” can be written as :

9

1)1(1

rtArt

AP

ImportantNote that in computing the interest it is customary to consider a 360-day year instead of a 365-day year. Thus 30 days will be considered as of an ordinary year and so on. The interest thus obtained is called “ordinary interest” but if it is based on 365 days it is called the “exact interest”.

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12

1

360

30

Application of the formula

Example:A credit union has issued a 3-year

loan of $5,000. Simple interest is charged at the rate of 10% per year. The principal plus interest is to be paid at the end of third year. Compute the interest for the 3-year period. What amount will be repaid at the end of the third year?

Solution: On board

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Simple DiscountIf “A” is the amount to be paid at maturity after a time “t” at the simple interest rate of “r percent” per annum. Then the simple interest “I” on maturity value “A” in time “t” is given by

To get the present “P” value we must subtract this from “A”.

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ArtI

)1( rtAP

ArtAP

Discounting Negotiable Instrument

A written promise to pay money at a certain specific date is called Negotiable Instrument.They are of two types,

Non interest bearingInterest bearing

The basic principles of discounting a bill of exchange or short term note at a bank or at any other party are the same as those of obtaining a loan from a bank which deducts interest in advance.

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Discounting non-Interest-Bearing Note

Example:After Khalid accepted a bill for Rs. 4,500.

Hanif discounted it at National Bank Karachi on April 15. The maturity date of the bill was May 15. How much did Hanif receive if the bill was discounted at 8%?

Solution:Period of discount = 30 days or t = 1/12 year.A = 4,500 & r = 8%the discounted value

P = A(1-rt) = Rs. 4,47014

Discounting Interest-Bearing Note

We follow the following two steps in discounting an interest-bearing note.

i. Find the maturity value from the face value of the note after adding the interest which would have been earned up to the maturity date at the given rate.

ii. Find the proceeds by discounting the maturity value obtained in step (i) at the discounting rate.

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ExampleMohsin had a note for Rs. 15,000 with an interest rate of 6%. The note was dated January 12, 1983 and the maturity date was 90 days after date. On January 27, 1983 he took the note to his bank which discounted it at a rate of 7%. How much did he receive?

Solution:Step-1: Find maturity value AStep-2: Discount the maturity value at 7%

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Equivalent Values of Different Debts and their Payments

Sometimes a situation arises when a single debt or a set of debts are to be paid on different dates by means of a single payment or a set of payments. To satisfy both the creditor and the debtor, the values of payments should be equivalent to the values of the original debts on a certain date called the comparison date.Due to interest, a sum of money has different values at different times. Therefore a comparison date should first be chosen to equate the sum of the values of the original debts with the values of the desired payments on the same date. This process will bring the different debts and the subsequent payments on the same footing.

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ExampleA man owes Rs. 800, Rs. 1,000, and Rs. 200 due in 30 days, 60 days and 90 days respectively. If the rate of interest is 6%, when will a single payment of Rs. 2,020 will discharge ail the three debts?

Solution:

Let the comparison date be 90 days from now. The given diagram illustrates the situation.

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Continue…

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Debts Values on comparison date

Rs. 800 800[1+(60/360)(0.06)] = Rs. 808.00

Rs.1,000 1,000[1+(30/360)(0.06)] = Rs. 1005.00

Rs. 200 = Rs. 200

Rs. 2,000 Rs. 2,013.00

Total Borrowed

Total on comparison date

Continue…

The amount Rs. 2,013 on the comparison date is less than Rs. 2,020, therefore it will further earn interest for some more days over and above the comparison date.Suppose these days be “x”.Therefore,

2013 {1+(x/360)(0.06)} = 2,020Solving for “x”, we get

x = 20.86x = 21 days

Thus total no. of days = 90 + 21 = 111 days

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Getting More Involved

SolvingPRACTICE SET 9, p # 110-111&PROBLEM SET 9, p # 112-113

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CHAPTER 10

MATHEMATICS

OFFINANCE-II

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Compound Interest• Compounding involves the calculation of interest

periodically over the life of the loan (or investment).• After each calculation the interest is added to the

principal.• Future calculations are on the adjusted principal (old

principal plus interest).• Compound interest is the interest on the principal plus

the interest of prior periods.• Future value, or compound amount, is the final amount

of the loan or investment at the end the last period.

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Lets seeHow $1 will grow if it is calculated for 4 years at 8% annually?

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Some term to understand• Compounded annually: Interest calculated on the

balance once a year.• Compounded semiannually: Interest calculated

on the balance every 6 months or every ½ years.• Compounded quarterly: Interest calculated on

the balance every 3 months or every ¼ years.• Compounded monthly: Interest calculated on the

balance each month.• Compounded daily: Interest calculated on the

balance each day.• Number of periods: Number of years multiplied by

number of times the interest is compounded per year.

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For ExampleIf we compound $1 for 4 years at 8% annually, semiannually, or quarterly, the following periods will result:Annually: 4 years * 1 = 4 periodsSemiannually: 4 years * 2 = 8 periodsQuarterly: 4 years * 4 = 16 periods

• Rate for each period: Annual interest rate divided by the number of times the interest is compounded per year. Compounding changes the interest rate for annual, semiannual, and quarterly periods as follows: 26

Continue…

Annually: 8% / 1 = 8%Semiannually: 8% / 2 = 4%Quarterly: 8% / 4 = 2%

Note:Both the number of periods (4)

and the interest rate (8%) for the annual example do not change. The rate and periods (not years) will always change unless the interest is compounded yearly.

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Compound Amount FormulaLet

P = Principal i = interest rate per compounding

periodn = number of compounding periods

(number of periods in which the principal has earned interest)

S = compound amountThe compound amount after one period is

S = P + iPS = P(1 + i) 28

Continue…

Similarly, the compound amount after three periods is given by:

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compound amount

after two periods

compound amount after one period

+interest earned

during the secondperiod

=

2)1(

)1)(1(

)]1([i1PS

iPS

iiPS

iPi

3)1( iPS

Continue…

Thus we have the following definition:If an amount of money “P” earns interest compounded at a rate of “i” percent per period, it will grow after “n” periods to the compound amount “S”, where

This equation is often referred as the compound interest formula.The compound interest is given by:

Compound interest = S - P30

niPS )1(

Examples1. Find out the compound amount and the

compound interest at the end of three years on a sum of Rs. 20,000 borrowed at 6% compounded annually.

2. If Rs. 3,000 are invested at 6% interest compounded semi-annually what would it amount to at the end of 8 years?

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Effective Interest Rates

Interest rates are typically stated as the annual percentages. The stated annual rate is usually referred to as the nominal rate.

When interest is compounded semiannually, quarterly, and monthly, the interest earned during a year is greater than if compounded annually.

When compounding is done more frequently than annually, an effective annual interest rate can be determined.

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Continue…

Definition: The effective interest rate is the interest rate compounded annually which is equivalent to a nominal rate compounded more frequently than annually. The two rates would be considered equivalent if both will result in the same compound amount.

Example:The nominal rate of 8% compounded quarterly is equivalent to the effective rate of interest 8.24%.

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Formula for finding the Effective rate of interest

11 mie

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Wheree = the effective ratei = interest rate per conversion periodm = number of conversion periods in one year

Formula for Finding the Equivalent Rates of Interest

??

?1

?1

rx

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Unknown nominal interest rate

# of times the compounding isrequired per year (for nominal rate)

# of periods fornominal rate

# of periods forthe given rate

Given interest rate

# of times the compounding isrequired per year (for given rate)

Example

At what nominal rate compounded quarterly will a principal accumulate to the same amount as at 8% compounded semi-annually?SOLUTION ON BOARD

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Depreciation by Reducing Balance Method

If ‘C’ is the original cost of a machinery and ‘T’ is the trade in or scrap value of the machinery after ‘n’ years of useful life and ‘r’ is the percentage rate of depreciation on the reduced balance each year.

Then,

Depreciation for 1st year = Cr

Residual value after 1st year = C – Cr = C(1-r)

Depreciation for the 2nd year = C(1-r)r

Residual value after 2nd year = C(1-r) – C(1-r)r = C(1-r)2

Continuing in this way we get:

Residual value after ‘n’ years = C(1-r)n 37

Continued….

Since ‘n’ is the useful life of the machinery when its residual value is ‘T’, therefore,

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n

n

n

n

C

Tr

C

Tr

C

Tr

TrC

1

1

)1(

)1(

Getting More Involved

Discussion on Practice Set 10-A, (P# 121/122)

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PRESENT VALUE(AT COMPOUND INTEREST)

The compound amount formula is given by:

A slight rearrangement of the formula gives:

This formula is called the Present value or discounting formula.

The factor or is called the present value factor or the discounting factor.

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niPS 1

n

n iSi

SP

11

ni11 ni 1

Examples

i. Find the present value of Rs. 4,814.07 due at the end of 8 years if money is worth 6% compounded semi-annually.

ii. What sum of money invested at 6% compounded annually will amount to Rs. 500 in 4 years?

iii. Find the present value of Rs. 4,958.54 due at the end of 8½ years if money is worth 6% compounded semi-annually.

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Application on Discounting Interest bearing and Non-interest

bearing Notes

Example:A non-interest bearing note of Rs. 3,000 is due in 5 years from now. If the note is discounted now at 6% compounded semi-annually, what will be the proceeds and the compound discount? (SOL ON BOARD)

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Example: (interest bearing note)

An interest bearing note of Rs. 5,000 dated January 1,1980 at 6% compounded quarterly for 10 years was discounted on January 1, 1984. what were the proceeds and the compound discount if the note was discounted at 8% compounded semi-annually?

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Examples

1. Mr. Ahmad owes Mr. Bashir Rs. 5,000 in three years and Rs. 10,000 in 5½ years. How much should Mr. Ahmad pay at the end of 4 years which may be acceptable to Mr. Bashir if money is worth 8% compounded semi-annually?

2. Mr.Zahir owes to Mr. Mohmood Rs. 4,000 due in 2 years and Rs. 5,000 due in 4 years. If he agree to pay Rs. 4,500 now, how much he will have to pay to Mr. Mahmood three years from now to settle his two debts, if money is worth 6% compounded semi-annually?

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Continued…

3. Mr. Shamim owes Rs.750 due in 3 years and Rs. 1,000 due in 8 years. He and his creditor agreed to settle the two debts by making two equal payments one 5 years and the other in 6 years. If the money is worth 6% compounded semi-annually, what should be the amount of each payment?

4. Mr. Mushtaq borrowed Rs. 500 in one year, Rs. 1500 due in two years and Rs. 2,000 in two and half years from Jamil. If money is worth 6% compounded semi-annually, when can Mr. Mushtaq can discharge all the debts by a single payment of Rs. 4,000?

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Continued…

Discussion onPractice Set 10-B, P#(127-

128)

&Problem Set 10, P# (128-

129)

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CHAPTER # 11

MATHEMATICS OF FINANCE – IIIANNUITIES

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DEFINITIONSAn annuity is a series of periodic payments

(usually equal in amounts).The payments are made at regular intervals

of time such as annually, semi-annually, quarterly or monthly.

Examples of annuities includeRegular deposits to a savings accountMonthly carMortgage Insurance paymentsPeriodic payments to a person from a retirement

fund

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Continue…

If the payments are made at the end of the payment periods the annuity is called an “ordinary annuity”

If the payments are made at the beginning of each interval the annuity is called “annuity due”

The time between two successive payment dates is called “payment period”.

The time between the beginning of the 1st payment & the end of the last payment period is called the “term of the annuity”

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Sum of an annuityLet

R= payment per periodi= interest rate per periodn= number of annuity payments (also number of periods)S= sum (future value) of the annuity after n periods (payments)

The sum of the annuity is given by:

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i

iRS

n 11

Example

1. Find the amount of an annuity of Rs. 500 payable at the end of each year for 10 years, if the interest rate is 6% compounded annually.

Finding R when S is Known2. A father at the time of birth of his daughter decides

to deposit a certain amount at the end of each year in the form of an annuity. He wants that the sum of Rs. 20,000 should be made available for meeting the expenses of his daughter’s marriage which he expects to be solemnized just after her 18th birthday. If the payments accumulate at 8% compounded annually, how much should he start depositing annually?

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Use of the annuity table

• In table 5 at book page 309 values of the sum of an ordinary annuity of Re.1 are given. We use the symbol read as “s angle n at i ” is used for the factor .

• To search an entry in the table we consult the table against i and n, and then multiply by the payment per period.

• The use of the table will be discussed in the solution of problems.

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ins i

i n 1)1(

Finding n when S is known

Example:How many semi-annual payments of Rs. 100 each at an account in the form of an ordinary annuity will accumulate Rs. 3,000 if the interest rate is 8%?

Finding i when S is knownExample:

An annuity of Rs. 300 payable at the end of each quarter amounts to Rs. 13,200 in 8 years. What is the nominal rate of interest if it is compounded quarterly?

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Formula for interpolating the value of i

• Some times we can’t find the required value of i directly from the annuity table. For example, if

By following along the row n = 32 in table 5 we don’t find a value exactly equal to 44. so we have to interpolate this value.If “a” is the nearest value of i whose “A” entry in the table is less than 44 “b” is the nearest value of i whose entry “B” in the table is greater than 44“x” is the required value of IThen , use

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4432 is

AB

A

ab

ax

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Present Value of an Ordinary Annuity

• The present value of an annuity is an amount of money today which is equivalent to a series of equal payments in the future.

• For exampleIf a loan has been made, we may be interested in determining the series of payments (annuity) necessary to repay the loan with interest.

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Finding the Present Value of an Annuity

LetR = amount of an annuityi = Interest rate per compounding

periodsn = number of annuity payments (also,

the number of compounding)P = Present Value of the annuity

Then,

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i

iRP

n)1(1

Using table

• Table – 6 (book page 311) gives the present value of an ordinary annuity of Rupee 1 per period. The factor

is denoted by the symbol ( a angle n at i ). Thus the general formula for the present value of an annuity becomes:

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i

i n)1(1

ina

i

iRaRP

n

in

)1(1

Examples

1. A loan of Rs. 94 is to be paid back in monthly installments the first one starting after one month of the starting of the loan. If the interest is charged at the rate of 24% per annum on the unpaid principal, what will be the amount of the monthly installment?

2. Find the present value of an annuity of Rs. 600 pay able at the end of each year for 15 years if the interest rate is 5% compounded annually.

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Finding R when P is known

• Mr. Ahmad borrows Rs. 21,000 from a bank to build a house with the condition that he would pay back the loan in the semi-annual equal installments in four years with interest rate at 6% compounded semi-annually. If the first payment is to start at the end of first six monthly period, what would be amount of each installment?

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Finding n when P is known

• Mr. Ashraf wants to deposit his savings of Rs. 50,000 in a bank which offers 8% interest compounded semi-annually so as to withdraw Rs.25,000 at the end of each six months from the date of deposit. How many withdrawals will he or his heir (in case of his death) be able to make before the entire amount is exhausted?

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Annuity Due• When the periodic payments of an annuity starts at

the beginning of an interval rather than at the end of interval the annuity is called an annuity due. Its term begins on the date of the first payment and ends on one interval after the last payment is made.

• The annuity due has a payment at the beginning of each interest period but none at the end of the term. Therefore the formula for calculating an annuity due is given as:

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1 1( ) ( 1)n i n iS due R s R R s 1(1 ) 1

( ) 1ni

S due Ri

or

Present Value of an annuity due

• For the present value of an annuity due, we find out the present value of (n-1) periods of an ordinary annuity and then add the 1st payment which has the same present value. The formula for calculating the present value of an annuity due is given by:

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1 1( ) ( 1)n i n iP due R a R R a 11 (1 )

( ) 1ni

P due Ri

or

Examples1. If Rs. 250 are deposited at the beginning

of each quarter in a fund which earns interest at the rate of 8% compounded quarterly what will it amount to after the end of the year?

2. Mrs. Ahmad bought a sewing machine by paying Rs. 50 each month for 10 months, beginning from now. If money is worth 12% compounded monthly, what was the selling price of the machine on cash payment basis?

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PERPETUITYAn annuity whose payments starts on certain date and

continues indefinitely is called perpetuity. As the payments continues for ever, it is impossible to compute the amount of the perpetuity but its present value can be determined easily.

The formula for calculating the present value of the perpetuity is given as:

where R is the size of periodic payment and i is the interest rate per period.

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RP

i

Examples

1. Pakistan Manufacturing Co. is expecting to pay Rs. 4.80 every 6 months on the share of its stocks. What is the present value of a share if money is worth 8% compounded semi-annually?

2. Find the present value of Karachi Toy Company share which is expected to earn Rs. 5.60 every six month, if money is worth 8% compounded quarterly.

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Discussion on Selected Exercises from

Practice sets 11-A (p# 139), 11-B(p#157-159) &

Problem set 11(p# 160-162)

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Chapter 16

Matrix Algebra

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Matrix

• A matrix is a rectangular array of numbers enclosed in brackets or in bold face parenthesis. Matrices are represented by capital letters such as A, B, C, X, and Y etc.

• Examples of matrices are:

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1 4

8 0A

5 1

3 3

5 4

B

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9C

Cont’d

A matrix is described by 1st stating its number of rows and then its number of columns. This description of a matrix is known as order of a matrix.

In the above examples matrix A has the order 2 x 2 (2 by 2), B has the order 3 x 2 (3 by 2), and the order of C is 2 x 1 (2 by 1).

Generally if there are m rows in a matrix and n columns, the order of the matrix would be m x n and we may call it as m x n matrix.

If m = n, the matrix is called a square matrix.

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General Form of m x n Matrix

• Generally an m x n matrix may be in the form given below:

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11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

n

n

n

m m m mn

a a a a

a a a a

a a a a

a a a a

Operation with Matrices(i) Addition / Subtraction

Matrices of the same order can be added/Subtracted.

While adding/subtracting two matrices of the same order we add/subtract their corresponding elements.

Given two matrices:

then

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11 12 13

21 22 23

31 32 33

a a a

A a a a

a a a

11 12 13

21 22 23

31 32 33

b b b

B b b b

b b b

&

11 11 12 12 13 13

21 21 22 22 23 23

31 31 32 32 33 33

a b a b a b

A B a b a b a b

a b a b a b

Cont’d

(ii) Multiplication

a) Multiplication of a matrix by a real number (Scalar)

When a matrix is multiplied by a real number, each element of the matrix is multiplied by that real number.

The product obtained is a matrix of the same order.

Example Let

then

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1 6 9

0 1 2

1 5 3

D

1 6 9

3. 3. 0 1 2

1 5 3

3*1 3*6 3*9

3*0 3*( 1) 3*2

3*1 3*5 3*3

3 18 27

0 3 6

3 15 9

D

b) Multiplication of a matrix by another matrix Multiplication of two matrices is only possible if the

number of columns in the first matrix is equal to the number of rows in the second matrix.

If this condition is not satisfied multiplication will not be possible.

If the order of the first matrix is m x n and the order of the second matrix is n x p multiplication will be possible and the order of the resultant matrix will be m x p.

To obtain any element in the product matrix, 1st determine the row and column (in which the element lies) in the product matrix.

Multiply the row of the first matrix with that column of the second matrix, this value will give us that element.

Further the method is explained in the following examples.

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Examples• If possible multiply the following matrices.

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2 5 3

7 8 4

0 2 8

1 5 6 9 3 5

4 1 1

1 3 5 70 8 1

2 4 6 83 4 2

5 3 1 74 6 5

9 7 6 2

A B

C D

M N

with

with

with