Chapter 3: Matrices and applications Test A Name:©John Wiley & Sons Australia, Ltd 1 Chapter 3: Matrices and applications Test A Name: _____ Simple familiar 1 What is the most important

© John Wiley & Sons Australia, Ltd 1

Chapter 3: Matrices and applications Test A Name: _____________________ Simple familiar 1 What is the most important revision tool? Answer, your Workbook!

2 State which of the following equations are non-linear. a) b) c) d)

Option B, , is not a linear equation because is raised to the 2nd power and

is raised to the power of one half

1

3 Identify which of the following matrix operations exhibits the right distributive law. a) b)

c)

d)

Answer … Who Cares? Option C exhibits the right distributive law.

.

1

4 The point is a valid solution for the linear system given by,

Determine the real value .

3

2 6 12 13x y z- + = -26 4 2x y z+ + = -

3 7 11x y z- + - =2 3 8 19x y z- - + =

26 4 2x y z+ + = -y

z

D O D+ =( )S T R ST SR+ = +

( )Q U E QE UE+ = +1 1F F FF I- -= =

( )Q U E QE UE+ = +

( )3, 2,c- -

2 3 4 165 6 7 34

2 4 11

x y zx y zx y z

+ + = -+ + = -

- - - =

c

( )( ) ( )

2 3 4 16Sub in the solution, 3, 2,

2 3 3 2 4 166 6 4 16

4 16 124

41

x y zc

cc

c

c

c

+ + = -

- -

- + - + = -

- + - + = -= - +-

=

= -

Maths Quest 12 Specialist Mathematics Units 3 & 4 for Queensland Chapter 3: Matrices and applications Test A


5 Refer to the matrices

and

.

to determine a unique solution for the linear system given by

4

6 Determine the restriction on the value for that would result in the following

system having no solution.

I don’t like this solution, but will leave it here for you to look at

I have a second and third solution following

A system with no solution has the property,

The two lines are coincident if

3

4 2 8 1,

3 4 3 2A B

- -é ù é ù= =ê ú ê ú- -ë û ë û

7 1118 24

C é ù= ê ú- -ë û

AX BX C- =

( )( ) ( ) ( )

( )( )

1 1

1

1

14 2 8 13 4 3 2

7 1118 24

2 31 1

AX BX CA B X C

A B A B X A B C

IX A B C

X A B C

X

X

- -

-

-

-

- =

- =

- - = -

= -

= -

æ ö- -é ù é ù= -ç ÷ê ú ê ú- -ë û ë ûè ø

é ù´ ê ú- -ë û

- -é ù= ê úë û

a

4 612 18 24x y ax y- =- =

2 2 2

1 1 1

a b ca b c= ¹

2 2

1 1

12 18 34 6

a ba b=

-= =-

2

1

3cc=

1

24 3

24388

c c

c

c

ca

=

¹

¹

\ ¹Þ ¹





These are linear functions!, hence for no solution they need to be parallel, but not colinear. Hence, they simply can’t be scalars of each other;

𝑐 ∈ 𝑅, 𝑏𝑢𝑡𝑐 ≠ 8

So that’s a Methods solution!



For a specialist solution we need to refer to what’s in the Syllabus … and that’s the 3 different solutions of a system of equations.

So let’s refer to RREF;

+ 4 612 18 |

𝐴242

Through a few Row operations, this becomes;

+4 60 0 |

𝐴8 − 𝐴2

So our third row is the problem … if we get three

zero’s we have a free variable, which gives us an Infinite number of solutions … that is, they are colinear!.

So we simply need for it NOT to be zero,

8 − 𝐴 ≠ 0 Hence we arrive at the only thing A can’t equal,

𝐴 ≠ 8

𝑐 ∈ 𝑅, 𝑏𝑢𝑡𝑐 ≠ 8

9 Determine a general solution for the variable matrix in the matrix equation

.

3

a

4 612 18 24x y ax y- =- =

a

4 612 18 24x y ax y- =- =

X

2 3AX BX C+ =

( )( ) ( ) ( ) ( )

( ) ( )( ) ( )

1 1

1

1

2 32 3

2 2 2 3

2 3

2 3

AX BX CA B X C

A B A B X A B C

IX A B C

X A B C

- -

-

-

+ =

+ =

+ + = +

= +

= +



10 Solve the following system of linear equations by using an inverse matrix that has been calculated using technology.

3

11 The Leslie matrix (L) below shows the fecundity rate of a female turkey population while N1 shows the population in the first year.

Calculate the total number of turkeys after six years, assuming there are equal numbers of male and female turkeys.

To find the population after six years use the result, N5 = L5 × N1

=

=

The total number of female turkeys will be 1630 so we can assume the total population will be 3260.

3

7 1545 6 5 1504 2 8 212

x y zx y zx y z

+ + = -+ - =

- - + = -

1

1

1 1 7 1545 6 5 1504 2 8 212

1 1 7 1 1 75 6 5 5 6 54 2 8 4 2 8

1 1 7 1545 6 5 1504 2 8 212

1 1 75 6 54 2 8

xyz

xyz

xyz

-

-

-é ù é ù é ùê ú ê ú ê ú- ´ =ê ú ê ú ê úê ú ê ú ê ú- - -ë û ë û ë û

é ù é ù é ùê ú ê ú ê ú- - ´ê ú ê ú ê úê ú ê ú ê ú- - - -ë û ë û ë û

-é ù é ùê ú ê ú= -ê ú ê úê ú ê ú- - -ë û ë û

é ù é ùê ú ê ú= -ê ú ê úê ú ê ú- -ë û ë û

1 154150212

7023

xyz

- -é ùê úê úê ú-ë û

é ù é ùê ú ê ú=ê ú ê úê ú ê ú-ë û ë û

7, 0, 23x y z\ = = = -

1

0.9 2.1 0.6 600.6 0 0 , 400 0.4 0 25

L Né ù é ùê ú ê ú= =ê ú ê úê ú ê úë û ë û

50.9 2.1 0.6 600.6 0 0 400 0.4 0 25

é ù é ùê ú ê úê ú ê úê ú ê úë û ë û

113639896

é ùê úê úê úë û



12 The following descriptions relate to the geometric and/or algebraic properties of a system of linear equations in three variables. Make a justified assertion on the type of solution set each system would produce based on the given information a) Two planes are distinct with parallel

normal vectors. The third plane is not parallel.

b) The upper triangular augmented

matrix is .

a) It is an inconsistent system; hence zero solutions exist. The planes intersect in lines as pairs.

b) The normals are coplanar and the planes intersect as a line. Infinitely many solutions can be defined parametrically. (What the?)

NAH … this is our answer to b) … Because row three are all zero’s, we have a free variable, which gives us an Infinite number of solutions. Clearly from Row 1 and Row 2 the equation are not scalars of each other, hence the 3 planes are NOT co-incident, hence the planes intersect at a line

3

13 Given,

and

Use technology to determine if the statement , is true.

The statement is false.

4

3

23

1

0

10 1

0

0

0 0

aa baé ù

ê úê úê úë û

4 9 72 4 101 2 0

A-é ùê ú= -ê úê ú-ë û9 8 53 6 20 2 4

B-é ùê ú= -ê úê úë û

4 3 3 4A B B A´ = ´

4 3

4 3

3 4

3 4

4 9 7 9 8 52 4 10 3 6 21 2 0 0 2 4

4625499 5619818 1057541343008 420320 87724887289 1078270 203497

9 8 5 4 9 73 6 2 2 4 100 2 4 1 2 0

2437273 5276

LHS A B

RHS B A

= ´

- -é ù é ùê ú ê ú= - ´ -ê ú ê úê ú ê ú-ë û ë û-é ùê ú= - -ê úê ú- -ë û

= ´

- -é ù é ùê ú ê ú= - ´ -ê ú ê úê ú ê ú-ë û ë û-

=685 5860997

1120859 2426877 2699397159182 344766 385166

LHS RHS

-é ùê ú-ê úê ú- -ë û

¹



14 A system of linear equations in three variables contains the lines . Determine if a solution exists for the simultaneous equations. If so, calculate the point of intersection and describe the geometric nature of the system by referring to any possible solution/s.

For , let

The following point exists on :

Verify whether lies on the second line.

The values for are not equal which indicates

the point does not lie on .

The two are parallel and distinct which means they will never intersect.

5

1 2 and l l

1 1

2 2

9 4 1: 2 3 4

2 7 4: 6 9 12

x y zl k

x y zl k

+ - += = =

-+ - +

= = =-

1

2

1 2 1 2

ˆˆ ˆ2 3 4ˆˆ ˆ6 9 12

13

d i j k

d i j k

d d d d

= - + +

= - + +

= ´ ® \

!

!

"! ! ! !

1 1

1

1

: 9 24 31 4

l x ky kz k

= - -= += - +

2 2

2

2

: 2 67 94 12

l x ky kz k

= - -= +

= - +

1l 1 0k =

1 : 9 2 0 94 3 0 41 4 0 1

l xyz

= - - ´ = -= + ´ == - + ´ = -

1l

( ) ( )1 1 1, , 9,4, 1x y z = - -

( )9,4, 1- -

2 2 2

2 2 2

2 2 2

2 6 7 9 4 129 2 6 4 7 9 1 4 12

7 3 16 9 4

x k y k z kk k k

k k k

= - - = + = - +- = - - = + - = - +

= = - =

2k

( )9,4, 1- - 2l


Chapter 3: Matrices and applications Test A Name: _____________________ Complex familiar



15 Consider the following system of linear equations to answer the following.

Solve the system of linear equations, expressing the solution in parametric form.

Describe how the solution can be interpreted geometrically.

The normal vectors are not parallel.

The three planes are coplanar, and not parallel. Geometrically, this could be the planes intersecting in a line or not at all.

The system has infinitely many solutions because the planes intercept in a line.

Let as it is the free variable.

Geometrically, the solution is the intersection line between three planes given by the parametric equations,

8

6 5 19 412 6 21

4 4 12 24

x y zx y zx y z

- - + =- - = -

- - + =

1 2

3

ˆ ˆˆ ˆ ˆ ˆ6 5 19 , 2 6 ,ˆˆ ˆ4 4 12

n i j k n i j k

n i j k

= - - + = - -

= - - +! !

!

( ) ( )( )

1 2 3

ˆ ˆˆ ˆ ˆ ˆ6 5 19 2 6

ˆˆ ˆ4 4 12

6 5 191 2 64 4 12

2 6 1 6 1 26 5 194 12 4 12 4 4

0

n n n

i j k i j k

i j k

× ´

= - - + × - -

´ - - +

- -= - -- -

- - - -= - - - +

- - - -

=

! ! !

[ ]

[ ]

6 5 19 41| 1 2 6 21

4 4 12 24

1 0 4 11| 0 1 1 5

0 0 0 0

A B

A B

é ù- -ê ú= - - -ê úê ú- -ë ûé ù- -ê ú= ê úê úë û

z t=

4 11 [1] , 5 [2]Given , [2] becomes

55

[1] becomes- 4 -11

11 4

x z y zz t

y ty t

x tx t

- = - + ==

+ == -

== - +

11 4 , 5 ,x t y t z t= - + = - =



16 Solve the following system of linear equation using Gaussian elimination and back substitution.

7

2 4 10 86 2 22

12 4 3 31

x y zx y zx y z

- + =+ + =

- + - = -1 2 1 3 2

2

3 2

3

2 , 3 , 2

13

6

1

2 4 10 86 1 2 2212 4 3 31

1 2 5 40 13 28 20 6 1 13

1 2 5 428 20 1

13 130 6 1 13

1 2 5 428 20 1

13 13181 1810 013 13

1 2 5 428 20 1

13 130 0 1 1

R R R R R

R

R R

R

÷ - +

÷

-

÷

-é ùê úê úê ú- - -ë û

-é ùê ú- -ê úê úë û

-é ùê ú- -ê úê úê úë û

é ùê ú-ê ú- -ê úê úê úê úë û

-é ùê ú- -ê úê úê úë û 81

13

1 [1]28 2 [2]

13 132 5 4 [3]

sub 1 in [2]:28 2

13 132 28

13 132

sub 2, 1 in [3]:2 2 5 1 44 4 53

z

y z

x y zz

y

y

yy z

xxx

=- -

+ =

- + ==

- -+ =

- -= -

== =

- ´ + ´ == + -=

( )The unique solution is given by,

3, 2, 1 or 3,2,1 .x y z= = =



17 Determine the equation of the line formed by the intersection between the following planes.

Give all possible answers in parametric form.

6

1

2

: 4 2 2 4: 3 3 8

p x y zp x y z

- + =- + + = -

( )

4 2 2 4 [1]3 3 8 [2]

[1] 2 [2]2 8 12 [3]

Let . Rewrite [3],2 8 12

12 8 6 42

[2] becomes,3 6 4 3 818 12 3 8

8 18 910 9

The parametric equation modelling the line of in

x y zx y z

x zz tx t

tx t

t y tt y t

y ty t

- + =- + + = -+ ´

- + = -=

- + = -- -

= = +-

- + + + = -

- - + + = -= - + += +

1

tersection is.6 4

: 10 9x t

l y tz t

= +ìï = +íï =î



18 In a round robin tennis competition, Sally’s two wins were against Abigail and Jenn. Abigail defeated Jenn. Melanie won all three of her games. Use dominance matrices and the formula

to rank the players.

Hence the dominance vector is given by,

.

The players are ranked Melanie, Sally, Abigail then Jenn.

5

23 2.2M M+

0 0 0 1 A1 0 1 11 0 0 10 0 0 0

A M S J

MM

SJ

é ùê úê ú=ê úê úë û

2

2

0 0 0 11 0 1 1

3 2.2 31 0 0 10 0 0 0

0 0 0 11 0 1 1

2.21 0 0 10 0 0 0

0 0 0 35.2 0 3 7.43 0 0 5.20 0 0 0

M M

é ùê úê ú+ =ê úê úë û

é ùê úê ú+ ´ê úê úë û


315.68.20

AM

VSJ



Chapter 3: Matrices and applications Test A Name: _____________________ Complex unfamiliar 19 The sum of three numbers is 6.

When double the first number and triple the second number are added to the third number the sum is 1. The last number is two less than the first. Develop a suitable matrix equation and hence calculate the values for each number.

The unknown numbers are 7,-6 and 5.

6 6 [1]2 3 1 [2]

2 [3 ]2 [3 ]1 1 1 6

| 2 3 1 11 0 1 2

1 1 1 6: 2 3 1 1

1 0 1 2

1 0 0 7| 0 1 0 6

0 0 1 5

x y zx y zz x ax z b

A B

rref

A B

+ + =+ + == -

- + = -

é ùê ú= ê úê ú- -ë ûìé ùïê úíê úïê ú- -ë ûîé ùê ú= -ê úê úë û



20 Two particles are following a straight-line path within the immediate vicinity of an electromagnetic field in the form of a flat plane modelled by the equation, . The vector equation of the first particle is given by,

Determine the vector equation of the second particle if it passes through

and the point in which the first particle intersects the plane.

Write in parametric form:

Substitute the parametric equation into the equation for the plane.

Determine the location on the line when .

The plane is intersected by the line at the point .

The second particle travels through and

.

The path of the second particle is given by the vector equation,

9

1 : 4 5 6 49p x y z+ - =

( ) ( )1ˆ ˆˆ ˆ ˆ ˆ: 4 7 2l i j k k i j k- + + + -

( )2, 4, 7P - - -

1l

4 , 1 2 , 7x k y k z k= + = - + = -

( ) ( ) ( )4 4 5 1 2 6 7 4916 4 5 10 42 6 49

20 49 3180204

k k kk k k

k

k

k

+ + - + - - =

+ - + - + == +

=

=

4k =

4 4 81 2 4 77 4 3

xyz

= + == - + ´ == - =

( )8,7,3Q

( )2, 4, 7P - - -

( )8,7,3Q

( ) ( )ˆ ˆˆ ˆ ˆ ˆ8 7 3 2 4 7

ˆˆ ˆ10 11 10

d q p

d i j k i j k

d i j k

= -

= + + - - - -

= + +

! ! !

!

!r a k d= +! ! !

( ) ( )

( ) ( )

ˆ ˆˆ ˆ ˆ ˆ2 4 7 10 11 10

ˆ ˆˆ ˆ ˆ ˆ8 7 3 10 11 10

r i j k k i j k

or

r i j k k i j k

= - - + - + + +

= + + + + +

!

!



21 It is known that is associated with the matrix,

.

Use a determinant approach to calculate all possible eigenvalues for .

8 1 9l =

1 04 7 00 0 2

cA

é ùê ú= ê úê úë û

A

( )

( ) ( )

3 2

3 2

1 04 7 00 0 2

0 det

1 0 0 00 det 4 7 0 0 0

0 0 2 0 0

1 00 det 4 7 1

0 0 2

0 4 8 10 23 14let 9

0 9 4 9 8 10 9 23 9 140 28 112

11228

4

cA

A I

c

c

c c

c cc

c

c

l

ll

l

ll

l

l l l ll

é ùê ú= ê úê úë û

= -

æ öé ù é ùç ÷ê ú ê ú= -ç ÷ê ú ê úç ÷ê ú ê úë û ë ûè øæ - öé ùç ÷ê ú= -ç ÷ê úç ÷ê ú-ë ûè ø

= - + - + - +=

= - + ´ - + - ´ +

= -

=

=

2 3

1 2 3

1 4 04 7 00 0 2

1 4 00 det 4 7 1

0 0 2

0 18 7 109, 2, 1

A

ll

l

l l ll l l

é ùê ú\ = ê úê úë ûæ - öé ùç ÷ê ú= -ç ÷ê úç ÷ê ú-ë ûè ø

= - - + -= = = -



22 The linear equations below form a consistent system with infinitely many solutions.

Calculate the unknown values, and .

The system has infinitely many solutions. The solution can be described by the parametric equations of a line.

For this to be true;

and

and

The directional vector of the line of intersection of the three planes is given by,

. As there are infinitely many solutions the free variable z will be defined as .

12

1

2

3 3 3

: 7 3 66 144: 5 4 2 20: 8 8

p x y zp x y zp x b y z d

- + + =+ + =+ + =

3b3d

1 2 3ˆ ˆ ˆ 0n n n× ´ =

1 2 2 3ˆ ˆ ˆ ˆn n n n´ = ´ 2 3ˆ ˆ0 n n= ×

1 2

ˆˆ ˆ

ˆ ˆ 7 3 665 4 2

3 66 7 66ˆ ˆ4 2 5 2

7 3ˆ5 4

ˆˆ ˆ258 344 43

i j kn n

i j

k

i j k

´ = -

-= -

-+

= - + -

ˆˆ ˆ258 344 43i j k- + -z t=

( )

7 3 66 144| 5 4 2 20

0 0 0 0

| :

1 0 6 12| 0 1 8 20

0 0 0 0

A B

ref A B

A B

é- ùê ú= ê úê úë û

é - - ùê ú= ê úê úë û

2

1

let From R : 8 20

20 8From R : 6 12

Let6 12

12 6

z ty ty t

x zz tx t

x t

=+ =\ = -- = -

=- = -

= - +



The line is defined by the parametric equations:

This solution must satisfy the third plane:

12 6 , 20 8 ,x t y t z t= - + = - =

( ) ( )

3 3 3

3 3

3 3 3

3

3

3

: 8 8Sub in 12 6 , 20 8 ,8 12 6 20 8 896 48 20 8 8

For this to satisfy the equation the terms containing the parameter must equal zero.

48 8 8 056 8 0

56

p x b y z dx t y t z tt b t t dt b tb t d

tt tb tt tb

b

+ + =

= - + = - =

- + + - + =

- + + - + =

- + =

- =

-=

3 3 3

3

3

78

Rewrite96 48 20 8 896 48 20 7 8 7 8

44

tt

t b tb t dt t t d

d

=-

- + + - + =

- + + ´ - ´ + =

=

3 : 8 7 8 44p x y z\ + + =

Documents

Chapter 3: Matrices and applications Test A Name:©John Wiley & Sons Australia, Ltd 1 Chapter 3: Matrices and applications Test A Name: _____ Simple familiar 1 What is the most important