3 Matrices

C

H

A

P

T

E

R

3

Matrices

Objectives

To be able to identify when two matrices areequal

To be able to add and subtract matrices of thesame dimensions To be able to perform multiplication of a matrix and ascalar

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

To be able to performmultiplicationon two suitable matrices To be able to find theinverseof a2×2matrix

To be able to find thedeterminantof a matrix

To be able to solvelinear simultaneous equationsin two unknowns using an inverse matrix

3.1

Introduction to matrices

Amatrixis a rectangular array of numbers. The numbers in the array are called the entries in the matrix.

The following are examples of matrices: 

 

−1 2 −3 4 5 6  

 [2 1 5 6]

  

√

2 3

0 0 1 √

2 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.

62

Example 1

Write down the dimensions of the following matrices.

a

1 1 2 2 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,CandDsell 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 of numbers. 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 that:

exporterAsold 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 exporterAare 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

IfAis a matrix,aijwill be used to denote the entry that occurs in rowiand columnjofA.

Thus a 3×4 matrix may be written:

A=

  

ForB, anm×nmatrix may be written:

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 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 modeljsold by showroomi.

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 matrices

1 2 3 4

and

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,Bareequal, and can be written asA=Bwhen: each has the same number of rows and the same number of columns they have the same number or element at corresponding positions.

For example, 

2 1 −1

0 1 3



₌



1+1 1 −1

64

Example 2

If matricesAandBare equal, find the values ofxandy.

A=

2 1

x 4

B=

2 1 −3 y

Solution

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 the following:

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 teams. Team D has 3 senior teams and 3 junior teams. Represent this information in a matrix.

Solution     

2 3 2 4 1 2 3 3     

Exercise

3A

1 Write down the dimensions of the following matrices.

Example1

a

1 2 3 4

b

2 1 −1

0 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 to

Example3

indicate whether a seat is vacant or occupied, write down a matrix that represents the situation when:

a only seats on the two diagonals are occupied b all seats are occupied.

3 If seating arrangements (as in Question 2) are represented by matrices, consider the matrix in which thei,jelement is 1 ifi = j, but 0 ifi = j. What seating arrangement does this matrix represent?

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

Example4

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 that could be equal, and write down the

Example2

values ofx,ywhich would make them equal.

a

3 2

,

0 x

,[0 x],[0 4 ]

b

4 7

1 −2

,

1 −2 4 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 matricesAandBare equal.

a A=

2 1 −1

0 1 3

B=

x 1 −1

0 1 y

b A=

x 2

B=

3 y

c A=[−3 x]B=[y 4] d A=

1 y 4 3

B=

1 −2

4 x

7 A section of a road map connecting townsA,B,C andDis shown. Construct the 4×4 matrix that shows the number of connecting roads between each pair of towns.

B

D

66

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 2 Express this data in a 5×3 matrix.

3.2

Addition, subtraction and multiplication

by a scalar

Addition 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 0 0 2

+

0 −3

4 1

=

1 −3

4 3

and

  

a11 a12 a21 a22 a31 a32  

+

  

b11 b12 b21 b22 b31 b32  

=

  

a11+b11 a12+b12 a21+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 −1

−4 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, thenkAis anm×nmatrix whose elements arektimes the

corresponding elements ofA. Thus:

3

2 −2

0 1

=

6 −6

0 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 ofnmatrices each equal toAisnA

(wherenis a natural number).

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

3 6

+

2 4

=

12 24

+

2 4

=

14 28

X−Y=

2 4

−

3 6

=

−1 −2

−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 −4

−2 8

,find matrixXsuch that 2A+X=B.

Solution

If 2A+X=B,thenX=B−2A

∴X=

0 −4 −2 8

−2×

3 2 −1 1

=

0−2×3 −4−2×2 −2−2× −1 8−2×1

=

−6 −8

0 6

68

Using a CAS calculator

Matrices are accessed through the Matrix Editor. Press the APPS key and select6:Data/Matrix Editorand then 3:New.

From the resulting menu select2:Matrix. Call this first matrixaand define it as a 2×2 matrix.

Press ENTER to obtain the edit screen. Note that the status in the top left of the screen is nowMat2×2. The entries are made in the usual way. This is defined

as matrixa. The matrixb=

3 6 5 6.5

is defined in

a similar way. Return to the Home screen.

The two matrices can be viewed by enteringaand thenb

in the entry line.

Entering matrices in the Home screen

This can be done row by row. For the matrix

3 6 6 7

enter

[[3,6][6,7]] and press ENTER. The matrix can be named by using STO. In the entry line [[3,6][6,7]]→a.

Addition, subtraction and multiplying

by a scalar

Exercise

3B

1 LetX=

1 −2

,Y=

3 0

,A=

1 −1

2 3

,B=

4 0 −1 2

FindX+Y,2X,4Y+X,X−Y,−3Aand−3A+B. Example6

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 3.1.)

a Given that the sales matrix for January is

3 6 2 4 2 1

, write down the sales matrix for

February.

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 −1

0 2

.

Find 2A,−3Aand−6A.

4 A,B,Carem×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 −3

4 1

Calculate:

a 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 matricesXandYsuch that

Example7

70

8 MatricesXandYshow the production of four modelsa,b,c,dat two automobile factories P,Qin 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.

3.3

Multiplication of matrices

Multiplication 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 3 4 2

andB=

5 1 6 3

Then AB=

1 3 4 2

5 1 6 3

=

1×5+3×6 1×1+3×3 4×5+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+1×2 6×1+3×4 6×3+3×2

=

9 17 18 24

Note thatAB=BA.

IfAis anm×nmatrix andBis ann×r matrix, then the productABis them×r matrix whose entries are determined as follows.

To find the entry in rowiand columnjofAB, single out rowiin matrixAand columnjin matrixB. Multiply the corresponding entries from the row and column and then add up the resulting products.

Example 8

ForA=

2 4

3 6

andB=

5 3

findAB.

Solution

Ais a 2×2 matrix andBis a 2×1 matrix. ThereforeABis defined. The matrixABis a 2×1 matrix.

AB=

2 4 3 6

5 3

=

2×5+4×3 3×5+6×3

=

22 33

Example 9

MatrixXshows the number of cars of modelsaandbbought by four dealers,A,B,CandD. MatrixYshows the cost in dollars of modelaand modelb.

FindXYand 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

Solution

a b

XY=

A B C D

    

3 1 2 2 1 4 1 1     

26 000 32 000

a b

4×2 2×1

The matrixXYis a 4×1 matrix.

XY=

    

3×26 000+1×32 000 2×26 000+2×32 000 1×26 000+4×32 000 1×26 000+1×32 000    

=

    

110 000 116 000 154 000 58 000

    

72

Example 10

ForA=

2 3 4 5 6 7

andB=

  

4 0 1 2 0 3

 

findAB.

Solution

Ais a 2×3 matrix andBis a 3×2 matrix. ThereforeABis a 2×2 matrix.

AB=

2 3 4

5 6 7

  

4 0

1 2

0 3

  

=

2×4+3×1+4×0 2×0+3×2+4×3 5×4+6×1+7×0 5×0+6×2+7×3

=

11 18

26 33

Using a CAS calculator

Multiplication ofA=

3 6

6 7

andB=

3 6

5 6.5

.

ABandBAare shown.

Exercise

3C

1 IfX=

2

−1

,Y=

1 3

,A=

1 −2

−1 3

,B=

3 2

1 1

,C=

2 1

1 1

,I=

1 0

0 1

,

Examples8,10

find the productsAX,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 in Question 1 defined?

AY,YA,XY,X2,CI,XI

b IfA=

2 0

0 0

andB=

0 0

−3 2

,findAB.

3 MatricesAandBare 2×2 matrices, andOis the zero 2×2 matrix. Is the following argument correct?

4 IfL=[2 −1],X= 2

−3 , findLXandXL.

5 AandBare bothm×nmatrices. AreABandBAdefined and, if so, how many rows and columns 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 the left-hand side is reversed?

7 Using the result of Question6, write down a pair of matricesA,Bsuch that

AB=BA=I, whereI=

1 0 0 1

.

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

9 It takes John five minutes to drink a milkshake that costs $2.50, and 12 minutes to eat a

Example9

banana split that 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.

10 The reading habits of five studentsA,B,C,DandEare shown in the first matrix below where the columnsp,q,r, andsrepresent 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 modeljsold from showroomi. Let the prices of the

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

Call the 3×1 matrix,C=

   c1 c2 c3  

the price matrix.

74

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 $u1,$u2and $u3respectively.

Form a new cost matrix.

C=

  

c1 u1 c2 u2 c3 u3   

FindSCand 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?

3.4

Identities, inverses and determinants

for 2

×

2 matrices

Identities

A 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 identityIexists.

For example, for 2×2 matrices I=

1 0 0 1

and for 3×3 matrices I=

  

1 0 0 0 1 0 0 0 1   

IfA=

2 3 1 4

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

appropriate multiplicative identity.

Inverses

Given a 2×2 matrixA, is there a matrixBsuch 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

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

B=

0.8 −0.6 −0.2 0.4

Bis said to be theinverseofA, asAB=BA=I.

LetAbe a 2×2 matrix withA=

a b c d

and letB=

x y u v

, whereBis 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,uandy,vrespectively. Taking thex,upair and eliminatingu, (ad−bc)x =d

Similarly, eliminatingx, (bc−ad)u=c

These two equations can be solved forxandurespectively, providedad−bc=0.

x = d

ad−bc andu= c cb−ad =

−c ad−bc

In a similar way it can be found that:

y= −b

ad−bc andv= −a cb−ad =

a ad−bc

Therefore the inverse=     

d ad−bc

−b ad−bc −c

ad−bc a ad−bc

    .

The inverse of a square matrixA, is denoted byA−1. The inverse is unique. ad−bchas a name, thedeterminantofA. This is denoted det(A).

For example,A=

a b c d

,det(A)=ad−bc.

A 2×2 matrix has an inverse only if det(A)=0.

A square matrix is said to beregularif its inverse exists. Those square matrices which do not have an inverse are calledsingularmatrices; for asingularmatrix det(A)=0.

Using a CAS calculator

The operation of matrix inverse is obtained by entering

a∧−1 in the entry line. The determinant is obtained through theMATHmenu, which is obtainable by pressing 2ND 5 and selecting4:Matrixand then the

76

Example 11

For the matrixA=

5 2

3 1

find:

a det(A) b A−1 Solution

a det(A)=5×1−2×3= −1 b A−1 = 1

−1

1 −2 −3 5

=

−1 2 3 −5

Example 12

For the matrixA=

3 2

1 6

find:

a det(A) b A−1 c X, ifAX=

5 6

7 2

d Y, ifYA=

5 6

7 2

Solution

a det(A)=3×6−2=16 b A−1 = 1

16

6 −2 −1 3

c AX=

5 6 7 2

Multiply both sides (from the left) byA−1.

A−1AX=A−1

5 6 7 2

∴IX=X= 1

16

6 −2 −1 3

5 6 7 2

= 1 16

16 30 16 0

=

1 2 1 0

d YA=

5 6 7 2

Multiply both sides (from the right) by

A−1_.

YAA−1 = 1

16

5 6 7 2

6 −2 −1 3

∴ YI=Y= 1

16

24 8 40 −8

∴ Y=

    3 2

1 2 5 2

−1 2

Exercise

3D

1 For the matricesA=

2 1 3 2

andB=

−2 −2

3 2

find:

Example11

a det(A) b A−1 _{c det(B) d B}−1

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

a

3 −1 4 −1

b

3 1 −2 4

c

1 0 0 k

d

cos −sin sin cos

3 IfA,Bare the regular matricesA=

2 1

0 −1

,B=

1 0 3 1

, findA−1,B−1.

Also findABand hence find, if possible, (AB)−1.

Also, fromA−1_,B−1_{, find the productsA}−1_B−1_andB−1_A−1_{. What do you notice?}

4 Let matrixA=

4 3 2 1

.

Example12

a FindA−1. b IfAX=

3 4 1 6

, findX. c IfYA=

3 4 1 6

, findY.

5 LetA=

3 2 1 6

,B=

4 −1

2 2

andC=

3 4 2 6

.

a FindXsuch thatAX+B=C. b FindYsuch thatYA+B=C.

6 IfAis a 2×2 matrix,a12=a21=0,a11=0,a22=0, then show thatAis regular and findA−1.

7 LetAbe a regular 2×2 matrix,Ba 2×2 matrix andAB=0. Show thatB=0.

8 Find all 2×2 matrices such thatA−1=A.

3.5

Solution of simultaneous equations

using matrices

Inverse matrices can be used to solve certain sets of simultaneous linear equations. Consider the equations

78

This can be written as

3 −2 5 −3

x y

=

5 9

IfA=

3 −2 5 −3

the determinant ofAis 3(−3)−5(−2)=1

which is not zero and soA−1exists.

A−1 =

−3 2 −5 3

Multiplying the matrix equation

3 −2 5 −3

x y

=

5 9

on the left hand side byA−1and using

the fact thatA−1A=Iyields the following:

A−1

A

x y

=A−1

5 9

∴ I

x y

=A−1

5 9

∴

x y

=

3 2

sinceA−1

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 parallel straight lines, a singular matrix results.

For example the system

x+2y=3 −2x−4y=6

has associated matrix equation

1 2

−2 −4 x y

=

3 6

Note that the determinant of

1 2 −2 −4

=1× −4−(−2×2)=0.

Example 13

If A=

2 −1

1 2

andK=

−1 2

, solve the systemAX=KwhereX=

x y

.

Solution

IfAX=K,thenX=A−1K

A−1K= 1

5

2 1 −1 2

×

−1 2

=

0 1

∴X=

0 1

Example 14

Solve the following simultaneous equations.

3x−2y=6 7x+4y=7

Solution

The matrix equation is

3 −2 7 4

x y

=

6 7

.

Let A=

3 −2 7 4

Then A−1 = 1

26

4 2 −7 3

and

x y

= 1 26

4 2 −7 3

6 7

= 1 26

38 −21

Using a CAS calculator

Enter

3−2 7 4

∧ (−1)∗

6 7

80

Exercise

3E

1 IfA=

3 −1 4 −1

, solve the systemAX=K, whereX=

x y

, and:

Example13

a K=

−1

2

b K=

−2

3

2 IfA=

3 1 −2 4

, solve the systemAX=K, where:

a K=

0 1

b K=

2 0

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

Example14

a −2x+4y=6 3x+y=1

b −x+2y= −1 −x+4y=2

c 2x+5y= −10 y=x+4

d 1.3x+2.7y= −1.2 4.6y−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 books and CDs. One child spends $120 and buys four books and four CDs. The other child buys three CDs and five books 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=3 4x−6y=6

a Write this system in matrix form asAX=K. b IsAa regular matrix?

Chapter summary

Amatrixis a rectangular array of numbers. Two matricesAandBare equal when:

r _{each has the same number of rows and the same number of columns, and} r _{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 anm×nmatrix whose elements arektimes the corresponding element ofA.

k

a b c d

=

ka kb kc kd

IfAis anm×nmatrix andBis ann×r matrix, then the productABis them×r matrix whose entries are determined as follows.

To find the entry in rowiand columnjofAB, single out rowiin matrixAand 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 ofAis the same as the number of rows ofB.

IfAandBare square matrices of the same dimension andAB=BA=IthenAis said to the inverse ofBandBis said to be the inverse ofA.

IfA=

a b c d

thenA−1 =

   

d ad−bc

−b ad−bc −c

ad−bc a ad−bc

   

det(A)=ad−bcis thedeterminantof matrixA.

A square matrix is said to beregularif 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 as

a b d e

x y

=

c

f

and

x y

=

a b d e

−1 c f

Review

82

Multiple-choice questions

1 The matrixA=

     1 0 2 −1 −2 3 3 0    

has dimension

A 8 B 4×2 C 2×4 D 1×4 E 3×4

2 IfA=

2 0 −1 3

andB=

1 −3 4 −1 −3 −1

thenA+B=

A

3 −3 −2 0

B

3 4 −2 2

C

−1 2 2 3

D

2 1

1 −3

E cannot be determined

3 IfC=

2 −3 1 1 0 −2

andD=

1 −3 1 2 3 −1

thenD−C=

A

1 0 0

−1 −3 −1

B

2 −6 4 −2 0 −4

C

−1 0 0 1 3 1

D

1 −6 0

1 3 1

E cannot be determined

4 IfM=

−4 0 −2 −6

then−M=

A

−4 0 −2 −6

B

0 −4 −6 −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 −2 −6 1

C

0 −4 −12 2

D

0 4

12 −2

E

0 2 6 −1

6 IfAandBare bothm×nmatrices, wherem =n, thenA+Bis an A m×nmatrix B m×mmatrix C n×nmatrix D 2m×2nmatrix E cannot be determined

7 IfPis anm×nmatrix, andQis an×pmatrix, the dimension of matrixQPis A 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 −1 1 −2

is

A −1 B

2 1

−1 −1

C

1 1

−1 −2

D

1 1 −1 2

E

10 IfM=

0 −2 −3 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=

0 2 3 4

andB=

1 3 0 5

, find:

a A+B b A−B c AB d det(A) e A−1

2 IfA=

1 0 2 3

andB=

−1 0

0 1

, find:

a (A+B)(A−B) b A2_−B2

3 Find all possible matricesAwhich satisfy the equation 3 4 6 8 A= 8 16 .

4 LetA=

1 2

3 −1

,B=[3 −1 2],C=

6 1

,D= 2 4andE=

   5 0 2   .

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

b EvaluateDAandA−1.

5 IfA=

1 −2 1 −5 1 2

,B=

  

1 −4 1 −6 3 −8  

andC=

1 2 3 4

,evaluateABandC−1_.

6 Find the 2×2 matrixAsuch thatA

1 2 3 4 = 5 6 12 14 .

7 IfA=

  

2 0 0 0 0 2 0 2 0  

,findA2 _{and henceA}−1_.

8 If

1 2 4 x

is a singular matrix, find the value ofx.

9 a IfM=

2 −1

1 3

,find the value of:

i MM=M2 ii MMM=M3 iii M−1

b Findxandy, given thatM

Review

84

Extended-response questions

1 A=

3 1

1 −4

,B=

2 −1

5 2

a Find:

i A+B ii A−B iii 2A+3B iv Csuch that3A+2C=B

b Find:

i AB ii A−1 iii Xsuch thatAX=B iv Ysuch thatYA=B

2 IfA=

  

1 −2 2 2 0 −1

1 3 4

  ,B=

  

−2 0 1 4 2 −2

1 3 3

 

andC=

  

2 0 2

3 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 following system of equations:

2x−3y =3 4x+y=5

i Write this system in matrix form, asAX=K. ii Find detAandA−1.

iii Solve the system of equations. iv Interpret your solution geometrically. b Consider the following system of equations:

2x+y=3 4x+2y =8