Cambridge Essentials General Math Specialist Chapter 1

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C H A P T E R

1MatricesObjectives

To be able to identify when two matrices are equal

To be able to add and subtract matrices of the same dimensions

To be able to perform multiplication of a matrix and a scalar

To be able to identify when the multiplication of two given matrices is possible

To be able to perform multiplicationon two suitable matrices

To be able to find the inverseof a2 2matrixTo be able to find the determinantof a matrix

To be able to solvelinear simultaneous equationsin two unknowns using an

inverse matrix

1.1 Introduction to matricesAmatrixis a rectangular array of numbers. The numbers in the array are called the entries in

the matrix.

The following are examples of matrices:

1 23 4

5 6

[2 1 5 6]

2 3

0 0 12 0

[5]

Matrices vary in size. The size, ordimension, of the matrix is described by specifying the

number of rows (horizontal lines) and columns (vertical lines) that occur in the matrix.

The dimensions of the above matrices are, in order:

3 2, 1 4, 3 3, 1 1.

The first number represents the number of rows and the second, the number of columns.

1

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2 Essential Advanced General Mathematics

Example 1

Write down the dimensions of the following matrices.

a

1 1 22 1 0

b

1

2

3

4

c 2 2 3

Solution

a 2 3 b 4 1 c 1 3

The use of matrices to store information is demonstrated by the following two examples.

Four exportersA,B,CandD sell televisions (t), CD players (c), refrigerators (r) and

washing machines (w). The sales in a particular month can be represented by a 4 4 array ofnumbers. This array of numbers is called a matrix.

r c w t

A

B

C

D

120 95 370 250

430 380 1000 900

60 50 150 100

200 100 470 50

row 1

row 2

row 3

row 4

column 1 column 2 column 3 column 4

From the matrix it can be seen:

ExporterA sold 120 refrigerators, 95 CD players, 370 washing machines and 250 televisions.ExporterBsold 430 refrigerators, 380 CD players, 1000 washing machines and 900 televisions.

The entries for the sales of refrigerators are made in column 1.

The entries for the sales of exporterA are made in row 1.

The diagram on the right represents a section of a road map.

The number of direct connecting roads between towns can be

represented in matrix form.

A B C D

A

B

C

D

0 2 1 1

2 0 1 0

1 1 0 0

1 0 0 0

B

A C

D

IfA is a matrix,aijwill be used to denote the entry that occurs in row i and columnjofA.

Thus a 3 4 matrix may be written

A =

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

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Chapter 1 Matrices 3

ForB, an m nmatrix

B =

b11 b12 . . . . . b1n

b21 b22 . . . . . b2n

. .

. .

. .

. .

. .

bm1 bm2 . . . . . bmn

Matrices provide a format for the storage of data. In this form the data is easily operated on.

Some graphics calculators have a built-in facility to operate on matrices and there are

computer packages which allow the manipulation of data in matrix form.

A car dealer sells three models of a certain make and his business operates through two

showrooms. Each month he summarises the number of each model sold by a sales

matrixS:

S =

s11 s12 s13

s21 s22 s23

, wheresi jis the number of cars of model jsold by showroom i.

So, for example,s12is the number of sales made by showroom 1, of model 2.

If in January, showroom 1 sold three, six and two cars of models 1, 2 and 3 respectively, and

showroom 2 sold four, two and one car(s) of models 1, 2 and 3 (in that order), the sales matrix

for January would be:

S =

3 6 2

4 2 1

A matrix is, then, a way of recording a set of numbers, arranged in a particular way. As in

Cartesian coordinates, the order of the numbers is significant, so that although the matrices1 2

3 4

,

3 4

1 2

have the same numbers and the same number of elements, they are different matrices (just as(2, 1), (1, 2) are coordinates of different points).

Two matricesA,B, areequal, and can be written as A = Bwhen

each has the same number of rows and the same number of columns

they have the same number or element at corresponding positions.

e.g.

2 1 1

0 1 3

=

1 + 1 1 1

1 1 1 62

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Example 2

If matricesA andB are equal, find the values ofxandy.

A = 2 1x 4

B = 2 13 ySolution

x = 3 andy = 4

Although a matrix is made from a set of numbers, it is important to think of a matrix as a

single entity, somewhat like a super number.

Example 3

There are four rows of seats of three seats each in a minibus. If 0 is used to indicate a seat is

vacant and 1 is used to indicate a seat is occupied, write down a matrix that represents

a the 1st and 3rd rows are occupied but the 2nd and 4th rows are vacant

b only the seat on the front left corner of the bus is occupied.

Solution

a 1 1 1

0 0 0

1 1 1

0 0 0

b 1 0 0

0 0 0

0 0 0

0 0 0

Example 4

There are four clubs in a local football league.

Team A has 2 senior teams and 3 junior teams

Team B has 2 senior teams and 4 junior teams

Team C has 1 senior team and 2 junior teamsTeam D has 3 senior teams and 3 junior teams

Represent this information in a matrix.

Solution

2 3

2 4

1 2

3 3

Note:rows represent teams A, B, C, D and columns represent the number

of senior and junior teams respectively.

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Exercise 1A

1 Write down the dimensions of the following matrices.Example 1

a 1 23 4

b 2 1 10 1 3

c [a b c d ] d

p

q

r

s

2 There are 25 seats arranged in five rows and five columns. If 0, 1 respectively are used toExample 3

indicate whether a seat is vacant or occupied, write down a matrix which represents the

situation when

a only seats on the two diagonals are occupied

b all seats are occupied.

3 If seating arrangements (as in2) are represented by matrices, consider the matrix in which

thei,jelement is 1 ifi = j, but 0 ifi = j. What seating arrangement does this matrixrepresent?

4 At a certain school there are 200 girls and 110 boys in Year 7, 180 girls and 117 boys inExample 4

Year 8, 135 and 98 respectively in Year 9, 110 and 89 in Year 10, 56 and 53 in Year 11 and

28 and 33 in Year 12. Summarise this information in matrix form.

5 From the following, select those pairs of matrices which could be equal, and write downExample 2

the values ofx,ywhich would make them equal.

a

32

,

0x

,[0 x],[0 4 ]

b

4 7

1 2

,

1 24 x

,

x 7

1 2

,[4 x 1 2]

c

2 x 4

1 10 3

,

y 0 4

1 10 3

,

2 0 4

1 10 3

6 In each of the following find the values of the pronumerals so that matricesA andB are

equal.

a A =

2 1 10 1 3

B =

x 1 10 1 y

b A =

x2

B =

3y

c A = [3 x] B = [y 4] d A =

1 y

4 3

B =

1 24 x

7 Asection of a road map connecting townsA,B,C

andD is shown. Construct the 4 4 matrix whichshows the number of connecting roads between

each pair of towns.

B

D

A C

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8 The statistics for the five members of a basketball team are recorded as follows.

Player A: points 21, rebounds 5, assists 5

Player B: points 8, rebounds 2, assists 3

Player C: points 4, rebounds 1, assists 1

Player D: points 14, rebounds 8, assists 60

Player E: points 0, rebounds 1, assists 2Express this data in a 5 3 matrix.

1.2 Addition, subtraction and multiplicationby a scalarAddition will be defined for two matricesonlywhen they have the same number of rows and

the same number of columns. In this case the sum of two matrices is found by adding

corresponding elements. For example,

1 00 2

+

0 34 1

=

1 34 3

and

a11 a12a21 a22

a31 a32

+

b11 b12b21 b22

b31 b32

=

a11 + b11 a12 + b12a21 + b21 a22 + b22

a31 + b31 a32 + b32

Subtraction is defined in a similar way. When the two matrices have the same number of rows

and the same number of columns the difference is found by subtracting corresponding

elements.

Example 5

Find

a

1 0

2 0

2 14 1

b

2 3

1 4

2 3

1 4

Solution

a

1 0

2 0

2 1

4 1

=

1 1

6

1

b

2 3

1 4

2 3

1 4

=

0 0

0 0

It is useful to definemultiplication of a matrix by a real number. IfAis anm nmatrix,andkis a real number, then kAis anm nmatrix whose elements are ktimes thecorresponding elements ofA. Thus

3

2 20 1

=

6 60 3

These definitions have the helpful consequence that if a matrix is added to itself, the result is

twice the matrix, i.e. A + A = 2A. Similarly the sum ofn matrices each equal toA is n A

(wherenis a natural number).Them nmatrix with all elements equal to zero is called thezero matrix.

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Example 6

Let X =

2

4

, Y =

3

6

, A =

2 0

1 2

, B =

5 0

2 4

FindX + Y, 2X, 4Y + X, X Y,3A,3A + B.

Solution

X + Y =

2

4

+

3

6

=

5

10

2X = 2

2

4

=

4

8

4Y + X = 4

36

+

24

=

1224

+

24

=

1428

X Y =

2

4

3

6

=

12

3A = 3

2 0

1 2

=

6 0

3 6

3A + B =6 0

3

6

+

5 0

2 4

=

1 0

1

2

Example 7

IfA =

3 2

1 1

andB =

0 42 8

, find matrices X such that 2A + X = B.

Solution

If 2A+

X=

B,

thenX=

B 2A X =

0 42 8

2

3 2

1 1

=

0 2 3 4 2 22 2 1 8 2 1

=

6 8

0 6

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Using the TI-Nspire2-by-2 matrices are easiest entered using

the 2-by-2 matrix template,t (/+r on

the clickpad), as shown.

Notice that there is also a template for

entering m by n matrices.

The matrix template can also be

obtained using/+b>Math Templates

To enter the matrix A=

3 6

6 7

, use the

Nav Pad to move between the entries of the2 by 2 matrix template andstore(/h)

the matrix as a.

Define the matrix B=

3 6

5 6.5

in a

similar way.

Entering matrices directly

To enter matrix A without using thetemplate, enter the matrix row by row as

[[3,6][6,7]] andstore(/h) the matrix

as a.

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Addition, subtraction and multiplication by a scalarOnce A andB are defined as above, A+ B,

AB andkA can easily be determined.

Using the Casio ClassPadMatrices are entered using the

2D CALC menu on the

k. Tap , enter the

numbers required then store

this as a variable (using VAR).

Calculations can be

performed as shown in the

screen at the far right.

Exercise 1B

1 Let X =

1

2

,Y =

3

0

,A =

1

1

2 3

,B =

4 0

1 2

FindX+Y, 2X, 4Y+X,X Y,3Aand3A+ B.

Example 6

2 Each showroom of a car dealer sells exactly twice as many cars of each model in February

as in January. (See example in section 1.1.)

a Given that the sales matrix for January is

3 6 2

4 2 1

, write down the sales matrix for

February.

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b If the sales matrices for January and March (with twice as many cars of each model

sold in February as January) had been

1 0 0

4 2 3

and

2 1 0

6 1 4

respectively, find the

sales matrix for the first quarter of the year.

c Find a matrix to represent the average monthly sales for the first three months.

3 LetA =

1 10 2

Find 2A, 3Aand6A.4 A,B,C are m nmatrices. Is it true that

a A + B = B + A b (A + B) + C = A + (B + C)?

5 A =

3 2

2 2

andB =

0 34 1

Calculatea 2A b 3B c 2A + 3B d 3B 2A

6 P =

1 0

0 3

,Q =

1 1

2 0

,R=

0 4

1 1

Calculate

a P + Q b P + 3Q c 2P Q + R

7 IfA =

3 1

1 4

andB =

0 10

2 17

,find matricesX andY such thatExample 7

2A 3X = Band 3A + 2Y = 2B.

8 MatricesXandYshow the production of four models a,b,c,dat two automobile factories

P,Q in successive weeks.

X =P

Q

a b c d 150 90 100 50

100 0 75 0

Y =

P

Q

a b c d 160 90 120 40

100 0 50 0

week 1 week 2

FindX + Yand write what this sum represents.

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1.3 Multiplication of matricesMultiplication of a matrix by a real number has been discussed in the previous section. The

definition for multiplication of matrices is less natural. The procedure for multiplying two

2 2 matrices is shown first.

Let A =

1 34 2

andB =

5 16 3

Then AB =

1 3

4 2

5 1

6 3

=

1 5 + 3 6 1 1 + 3 34 5 + 2 6 4 1 + 2 3

=

23 10

32 10

and BA =

5 1

6 3

1 3

4 2

=

5 1 + 1 4 5 3 + 1 26 1 + 3 4 6 3 + 3 2

=

9 17

18 24

Note thatAB

= BA.

IfA is anm nmatrix andB is ann rmatrix, then the product ABis them r matrixwhose entries are determined as follows.

To find the entry in rowi and columnjofABsingle out rowi in matrixA and columnjin

matrixB. Multiply the corresponding entries from the row and column and then add up the

resulting products.

Note:The productAB is defined only if the number of columns ofA is the same as the number

of rows ofB.

Example 8

ForA =

2 4

3 6

andB =

5

3

findAB.

Solution

Ais a 2 2 matrix andB is a 2 1 matrix. ThereforeAB is defined.The matrixABis a 2 1 matrix.

AB =

2 4

3 6

5

3

=

2 5 + 4 33 5 + 6 3

=

22

33

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Example 9

MatrixX shows the number of cars of modelsaandb bought by four dealers, A,B,CandD.

MatrixY shows the cost in dollars of model aand modelb.

FindXY and explain what it represents.

a b

X =

A

B

C

D

3 1

2 2

1 4

1 1

Y =

26 000

32 000

a

b

Solutiona b

XY =

A

BC

D

3 1

2 21 4

1 1

26 00032 000

ab

4 2 2 1

The matrixXY is a 4 1 matrix

XY =

3 26 000 + 1 32 0002 26 000 + 2 32 0001

26 000 + 4

32 000

1 26 000 + 1 32 000

=

110 000

116 000

154 000

58 000

The matrixXYshows dealerA spent $110 000, dealerB spent $116 000, dealerC

spent $154 000 and dealerD spent $58 000.

Example 10

ForA =

2 3 4

5 6 7

andB =

4 0

1 2

0 3

findAB.

Solution

Ais a 2 3 matrix andB is a 3 2 matrix. ThereforeABis a 2 2 matrix.

AB =

2 3 4

5 6 7

4 01 20 3

=

2 4 + 3 1 + 4 0 2 0 + 3 2 + 4 35

4 + 6

1 + 7

0 5

0 + 6

2 + 7

3

=

11 18

26 33

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Exercise 1C

1 IfX =

2

1

, Y =

1

3

, A =

1 21 3

, B =

3 2

1 1

, C =

2 1

1 1

, I =

1 0

0 1

,

Examples8,10

find the products AX, BX, AY, IX, AC, CA, (AC)X, C(BX), AI, IB, AB, BA,

A2, B2, A(CA) andA2C.

2 a Are the following products, of matrices given in1, defined?

AY, YA, XY, X2, CI, XI

b IfA =

2 0

0 0

andB =

0 0

3 2

, findAB.

3 The matrices AandB are 2

2 matrices, andOis the zero 2

2 matrix. Is the following

argument correct?

IfAB = O, andA = O, thenB = O.

4 IfL = [2 1], X =

2

3

, findLXandXL.

5 AandB are bothm nmatrices. AreABandBAdefined and, if so, how many rows andcolumns do they have?

6 Suppose a b

c d

d b

c a=

1 0

0 1.

Show thatad bc = 1. What is the product matrix if the order of multiplication on theleft-hand side is reversed?

7 Using the result of6, write down a pair of matrices A,B such thatAB = BA = Iwhere

I =

1 0

0 1

.

8 Select any three 2 2 matricesA,B andC.CalculateA(B + C), AB + ACand (B + C)A.

9 It takes John five minutes to drink a milk shake which costs $2.50, and twelve minutes toExample 9

eat a banana split which costs $3.00.

Calculate the product

5 12

2.50 3.00

1

2

and interpret the result in milk bar economics.

Suppose two friends join John.

Calculate

5 12

2.50 3.00

1 2 0

2 1 1

and interpret the result.

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10 The reading habits of five studentsA,B,C,D andEare shown in the first matrix below

where the columnsp,q,r, ands represent four weekly magazines. The second matrix

shows the cost in dollars of each magazine. Find the product of the two matrices and

interpret the result.

p q r s

A

B

C

D

E

0 0 1 1

1 0 1 1

1 0 0 0

1 1 1 1

0 1 0 1

p

q

r

s

2.00

3.00

2.50

3.50

11 LetS =

s11 s12 s13

s21 s22 s23

be the sales matrix for two showrooms selling three models of

cars. Heresijis the number of cars of model jsold from showroomi. Let the prices of the

three models of cars be $c1, $c2, $c3.

Call the 3 1 matrix,C =

c1c2

c3

the price matrix.

a FindSC. b What is the practical meaning ofSC?

c Suppose the car dealer sells both new and used cars and the price of two-year-old used

cars for the three models is $u1, $u2 and $u3,respectively.

Form a new cost matrix

C =

c1 u1c2 u2c3 u3

FindSC and state its meaning.

d Suppose the car dealer makes 30% profit on his selling of new cars and 25% on used

cars.

IfV =

0.3 0

0 0.25

, what is the meaning ofCV?

1.4 Identities, inverses and determinantsfor 2 2 matrices

IdentitiesA matrix with the same number of rows and columns is called a square matrix. For square

matrices of a given dimension, e.g. 2 2, a multiplicative identity I exists.

For example, for 2 2 matrices I =

1 0

0 1

and for 3 3 matrices I =1 0 00 1 0

0 0 1

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IfA =

2 3

1 4

, AI = IA = A, and this result holds for any square matrix multiplied by the

appropriate multiplicative identity.

InversesGiven a 2 2 matrixA, is there a matrix B such thatAB = BA = I?

LetB =

x y

u v

andA =

2 3

1 4

ThenAB = Iimplies

2 3

1 4

x y

u v

=

1 0

0 1

i.e.

2x + 3u 2y + 3v

x + 4u y + 4v

=

1 0

0 1

2x + 3u = 1 and 2y + 3v = 0

x + 4u = 0 y + 4v = 1

These simultaneous equations can be solved to findx,u,y, andv and henceB.

B =

0.8 0.60.2 0.4

Bis said to be theinverseofA as AB = BA = I.

LetA be a 2 2 matrix withA =

a b

c d

and letB =

x y

u v

whereB is the inverse ofA.

ThenAB = I. In full this is written

ax + bu ay + bv

cx + du cy + dv

=

1 0

0 1

Hence ax + bu = 1 ay + bv = 0

cx + du = 0 cy + dv = 1

which form two pairs of simultaneous equations, forx,u andy,vrespectively.

Taking thex , upair and eliminatingu , (ad bc)x = dSimilarly, eliminating x , (bc ad)u = cThese two equations can be solved forx andurespectively providedad

bc

= 0

i.e. x =d

ad bc andu =c

cb ad =c

ad bcIn a similar way it can be found that

y =b

ad bc andv =a

cb ad =a

ad bc

Therefore the inverse =

d

ad bcb

ad bcc

ad

bc

a

ad

bc

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The inverse of a square matrix A, is denoted byA1. The inverse is unique. ad bchas a

name, thedeterminantofA. This is denoted det(A).

i.e.forA =

a b

c d

, det(A) = ad bc

A 2 2 matrix has an inverse only if det(A) = 0A square matrix is said to be regularif its inverse exists. Those square matrices which do

not have an inverse are calledsingularmatrices; i.e. for a singularmatrix det(A) = 0.

Using the TI-NspireThe operation of matrix inverse is obtained

by raising the matrix to the power of1.

The determinantcommand

(b>Matrices & Vectors>Determinant)is used as shown.

(ais the matrixA =

3 6

6 7

defined

on page 8.)

Hint:you can also type in det(a)

Using the Casio ClassPadThe operation of matrix inverse is obtained

by entering A1 in the entry line.

The determinant is obtained by entering

and highlighting A and tappingInteractive,

Matrix-Calc, det.

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Example 11

For the matrixA =

5 2

3 1

find

a det(A) b A1

Solution

a det(A) = 5 1 2 3 = 1 b A1 = 11

1 23 5

=

1 2

3 5

Example 12

For the matrixA =

3 2

1 6

find

a det(A) b A1 c XifAX =

5 6

7 2

d YifYA =

5 6

7 2

Solution

a det(A) = 3 6 2 = 16 b A1 = 116

6 21 3

c AX = 5 67 2

Multiply both sides (from the left) by A1.

A1AX = A1

5 6

7 2

IX = X =1

16

6 21 3

5 6

7 2

=1

16 16 30

16 0=

1 2

1 0

d YA =

5 6

7 2

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Multiply both sides (from the right) by A1

YAA1 =1

16

5 6

7 2

6 21 3

YI = Y =1

1624 8

40 8

Y =

3

2

1

25

2

12

Exercise 1D

1 For the matricesA = 2 13 2

andB = 2 23 2

findExample 11a det(A) b A1 c det(B) d B1

2 Find the inverse of the following regular matrices (is any real number,kis any non-zero

real number).

a

3 14 1

b

3 1

2 4

c

1 0

0 k

d

cos sin sin cos

3 IfA,B are the regular matrices A = 2 1

0

1, B = 1 0

3 1, findA1

, B1.

Also findAB and hence find, if possible, (AB)1.

Also find fromA1, B1, the products A1B1 andB1A1. What do you notice?

4 For the matrixA =

4 3

2 1

Example 12

a findA1 b ifAX =

3 4

1 6

, findX c ifYA =

3 4

1 6

, findY.

5 IfA = 3 2

1 6, B = 4 12 2 andC =

3 4

2 6, finda Xsuch thatAX + B = C b Ysuch thatYA + B = C

6 IfA is a 2 2 matrix,a12 = a21 = 0, a11= 0, a22= 0, then show thatA is regular andfindA1.

7 LetA be a regular 2 2 matrix,B a 2 2 matrix andAB = 0. Show thatB = 0.

8 Find all 2 2 matrices such that A1 = A.

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1.5 Solution of simultaneous equationsusing matricesInverse matrices can be used to solve certain sets of simultaneous linear equations. Consider

the equations

3x 2y = 55x 3y = 9

This can be written as3 25 3

x

y

=

5

9

IfA =

3 25 3

the determinant ofA is 3(3) 5(2) = 1

which is not zero and soA1 exists.

A1 =

3 25 3

Multiplying the matrix equation

3 25 3

x

y

=

5

9

on the left hand side byA1 and using

the fact thatA1A = Iyields the following:

A1 Ax

y = A1 5

9 I

x

y

= A1

5

9

x

y

=

3

2

sinceA1

5

9

=

3

2

This is the solution to the simultaneous equations.

Check by substitutingx = 3,y = 2 in the equations.

When dealing with simultaneous linear equations in two variables which represent parallelstraight lines, a singular matrix results.

For example the system

x + 2y = 3

2x 4y = 6has associated matrix equation

1 2

2 4

x

y

=

3

6

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Note that the determinant of

1 2

2 4

= 1 4 (2 2) = 0.

There is no unique solution to the system of equations.

Example 13

If A =

2 11 2

andK=

1

2

, solve the system AX = KwhereX =

x

y

.

Solution

IfAX = K, then X = A1K

A1K=1

5

2 1

1 2

1

2

=

0

1

X =0

1

Example 14

Solve the following simultaneous equations.

3x 2y = 67x + 4y = 7

Solution

The matrix equation is

3 27 4

x

y

=

6

7

Let A =

3 27 4

Then A1 =1

26

4 2

7 3

and

x

y

=

1

26

4 2

7 3

6

7

=

1

26

38

21

Since any linear system ofn equations inn unknowns can be written as

AX = K, whereA is an n nmatrix,X =

x1

x2

.

.

xn

andK=

k1

k2

.

.

kn

,

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this method can be applied more generally when A is regular. In fact, as shown, an expression

for the solution can be written at once. MultiplyAX, andK, on the left by A1, and

A1(AX) = A1KandA1(AX) = (A1A)X = IX = X.

HenceX = A1K, which is a formula for the solution of the system. Of course it depends

on the inverseA1 existing, but onceA1 is found then equations of the form AX = Kcan be

solved for all possiblen 1 matricesK.Again this process can be completed using a calculator as long as matricesA andKhave

been entered onto the calculator.

Example 15

Consider the system of five equations in five unknowns.

2a + 3b c + d+ 2e = 9a + b

c = 4

a + 2d 3e = 4b + 2c d+ e = 6a b + d 2e = 0

Use a graphics calculator to solve fora,b,c,dande.

Solution

Enter 5 5 matrixA and 5 1 matrixB into the graphics calculator.

A =

2 3 1 1 21 1 1 0 01 0 0 2 30 1 2 1 11 1 0 1 2

B =

9

44

60

ThenA1B =

4

923

91

7

9

23

a =4

9, b =

23

9 , c = 1, d= 7

9 ande = 2

3

It should be noted that just as for two equations in two unknowns, there is a geometric

interpretation for three equations in three unknowns. There is only a unique solution if the

equations represent three planes intersecting at a point.

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Exercise 1E

1 IfA =

3 14 1

, solve the systemAX = KwhereX =

x

y

, andExample 13

a K=

12

b K=

23

2 IfA =

3 1

2 4

, solve the systemAX = Kwhere

a K=

0

1

b K=

2

0

3 Use matrices to solve the following pairs of simultaneous equations.Example 14

a 2x + 4y = 63x + y = 1

b x + 2y = 1x + 4y = 2

c 2x + 5y = 10y = x + 4

d 1.3x + 2.7y = 1.24.6y 3.5x = 11.4

4 Use matrices to find the point of intersection of the lines given by the equations

2x 3y = 7 and 3x + y = 5.

5 Two children spend their pocket money buying some books and some CDs. One child

spends $120 and buys four books and four CDs. The other child buys three CDs and fivebooks and spends $114. Set up a system of simultaneous equations and use matrices to find

the cost of a single book and a single CD.

6 Consider the system 2x 3y = 34x 6y = 6

a Write this system in matrix form, asAX = K.

b IsA a regular matrix?

c Can any solutions be found for this system?

d How many pairs does the solution set contain?

7 Consider the system of four equations in four unknowns.Example 15

p + q rs = 5r+s = 1

2p q + 2r= 2p q +s = 0

Use a graphics calculator to solve forp,q,rands.

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Chapter summary

Amatrixis a rectangular array of numbers.

Two matricesA andB are equal when: each has the same number of rows and the same number of columns, and they have the same number or element at corresponding positions.

The size ordimensionof a matrix is described by specifying the number of rows ( m) and the

number of columns (n). The dimension is writtenm n.Addition will be defined for two matrices only when they have the same dimension. The sum

is found by adding corresponding elements.a b

c d

+

e f

g h

=

a + e b + f

c + g d+ h

Subtraction is defined in a similar way.

IfAis anm nmatrix andkis a real number, kAis defined to be an m nmatrix whoseelements arektimes the corresponding element ofA.

k

a b

c d

=

ka kb

kc kd

IfAis anm nmatrix andB is ann rmatrix, then the productABis them rmatrixwhose entries are determined as follows.

To find the entry in rowi and columnjofAB, single out rowi in matrixA and columnjin

matrixB. Multiply the corresponding entries from the row and column and then add up the

resulting products.

The productABis defined only if the number of columns ofA is the same as the number

of rows ofB.

IfAandB are square matrices of the same dimension andAB = BA = IthenA is said to

the inverse ofB andB is said to be the inverse ofA.

IfA =

a b

c d

thenA1 =

d

ad bcb

ad bcc

ad

bc

a

ad

bc

det(A) = ad bcis thedeterminantof matrixA.A square matrix is said to be regularif its inverse exists. Those square matrices which do

not have an inverse are calledsingularmatrices.

Simultaneous equations can be solved using inverse matrices, for example

ax + by = c

d x + ey = f

can be written asa b

d ex

y=c

f

andx

y=a b

d e1 c

f

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Multiple-choice questions

1 The matrixA =

1 0

2 1

2 3

3 0

has dimensionA 8 B 4 2 C 2 4 D 1 4 E 3 4

2 IfA =

2 0

1 3

andB =

1 3 41 3 1

thenA + B =

A

3 32 0

B

3 4

2 2

C

1 2

2 3

D

2 1

1 3

E Cannot be determined

3 IfC= 2 3 11 0 2 andD =

1

3 1

2 3 1 thenD C =A

1 0 0

1 3 1

B

2 6 42 0 4

C

1 0 0

1 3 1

D

1 6 01 3 1

E Cannot be determined

4 IfM =

4 02 6

then M =

A

4 02 6

B

0 46 2

C

4 0

2 6

D

0 4

6 2

E

4 0

2 6

5 IfM =

0 2

3 1

andN =

0 4

3 0

then 2M 2N =

A

0 0

9 2

B

0 26 1

C

0 412 2

D

0 4

12 2

E

0 2

6 1

6 IfA andB are bothm nmatrices, wherem= n, thenA + Bis anA m nmatrix B m m matrix C n nmatrixD 2m 2nmatrix E Cannot be determined

7 IfP is anm nmatrix, andQis an pmatrix, the dimension of matrixQP isA n n B m p C n p D m n E Cannot be determined

8 The determinant of matrixA =

2 2

1 1

is

A 4 B 0 C 4 D 1 E 2

9 The inverse of matrixA =

1 11 2

is

A 1 B 2 11 1 C 1 11 2 D 1 11 2 E 2 11 1

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10 IfM =

0 23 1

andN =

0 2

3 1

thenNM =

A 0 4

9 1 B

4 2

2 8 C

0 4

9 1 D

6 2

3 5 E

6 2

3 5

Short-answer questions (technology-free)

1 IfA =

1 0

2 3

andB =

1 0

0 1

, find

a (A + B)(A B) b A2 B2

2 Find all possible matricesA which satisfy the equation 3 46 8

A = 816.

3 LetA =

1 2

3 1

, B = [3 1 2], C =

6

1

, D =

2 4

andE =

50

2

.

a State whether or not each of the following products exist:AB,AC,CD,BE

b EvaluateDAandA1.

4 IfA =

1 2 1

5 1 2

, B =

1 41 6

3 8

andC =

1 2

3 4

,evaluateABandC1.

5 Find the 2 2 matrixA such thatA

1 2

3 4

=

5 6

12 14

6 IfA =

2 0 00 0 2

0 2 0

,findA2 and henceA1.

7 If

1 2

4 x

is a singular matrix, find the value ofx.

8 a IfM =

2 11 3

,find the value of

i MM = M2 ii MMM = M3 iii M1

b Findx andy given thatM

x

y

=

3

5

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Extended-response questions

1 A =

3 1

1 4

, B =

2 15 2

a Findi A + B ii A B iii 2A + 3B iv Csuch that3A + 2C = B

b Find

i AB ii A1 iii Xsuch thatAX = B iv Ysuch thatYA = B

2 IfA =

1 2 22 0 1

1 3 4

, B =

2 0 14 2 2

1 3 3

andC

2 0 23 0 1

1 3 1

, find

a AB b AC c BC

d Xsuch thatAX = C e Ysuch thatYA = B

f Xsuch thatAXC = CB g Ysuch thatCYA = BA

3 a Consider the system of equations

2x 3y = 34x + y = 5

i Writethissystem in matrix form, asAX = K.

ii Find detAandA1.

iii Solve the system of equations.

iv Interpret your solution geometrically.

b Consider the system of equations

2x + y = 3

4x + 2y = 8

i Write this system in matrix form, asAX = K.

ii Find detAand explain whyA1 does not exist.

c Interpret your findings in partb geometrically.

4 The final grades for Physics and Chemistry are made up of three components, tests,

practical work and exams. Marks out of 100 are awarded in each component each

semester.

Wendy scored the following marks in each of the three components for Physics.

Semester 1: tests 79, practical work 78, exam 80


a Represent this information in a 2 3 matrix.To calculate the final grade for each semester the three components are

weighted so that tests are worth 20%. Practical work is worth 30% and the exam is worth

50%.(contd)

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b Represent this information in a 3 1 matrix.c Calculate Wendys final grade for Physics in each semester.

Wendy also scored the following marks in each of the three components for Chemistry.


Semester 2: tests 81, practical work 80, exam 70d Calculate Wendys final grade for Chemistry in each semester.

Students who gain an aggregate score for Physics and Chemistry of 320 or more over

the two semesters are awarded a Certificate of Merit in Science.

e Will Wendy be awarded a Certificate of Merit in Science?

She asks her teacher to remark her Semester 2 Chemistry Exam hoping that she will

gain the necessary marks to be awarded a Certificate of Merit.

f How many extra marks does she need?

5 A company runs Computing classes and employs full-time and part-time teaching staff as

well as technical support staff, cleaners and catering staff. The number of staff employed

depends on demand from term to term.

In one year they employed the following teaching staff:

Term 1: full-time 10, part-time 2




a Represent this information in a 4 2 matrix.Full-time teachers are paid $70 per hour and part-time teachers are paid $60 per hour.

b Represent this information in a 2 1 matrix.c Calculate the cost per hour to the company for teaching staff for each term.

In the same year they also employed the following support staff

Term 1: technical staff 2, catering staff 2, cleaning staff 1.




d Represent this information in a 4 3 matrix.Technical staff are paid $60 per hour, catering staff $55 per hour and cleaners are paid

$40 per hour.

e Represent this information in a 3 1 matrix.f Calculate the cost per hour to the company for support staff for each term.

g Calculate the total cost per hour to the company for teaching and support staff for each

term.

Documents

Cambridge Essentials General Math Specialist Chapter 1