2012 204 Matrices

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NAME : ________________________

Form : _______________matrices_21

1 (a) Find inverse of the matrix

41

62.

(b) Hence, sin! matrices, ca"c"ate the va"e

of x and of y that satisf# the fo""o$in!sim"taneos "inear e%ations:

=

1

0

41

62

m

k

(a) &

−

−21

64

2

1 (b)k = –3,

m = 1'

Answer :

2 iven that matrix

=

k G

1

34.

(i) Find the va"e of , if G does not have

an inverse matrix.(ii) iven that * 2,

(a) Find inverse matrix of G.(b) Hence, sin! matrices, ca"c"ate

the va"e of d and of e that satisf#thefo""o$in! matrix e%ations:

=

4

21

1

34

e

d

k

(i) *+- (ii) (a)

−

−41

32

5

1

(b) d = 6, e

= –1

Answer :

1

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+ M is a 2 x 2 matrix $here

=

−

−

10

01

45

23 M .

(a) Find the matrix M .

(b) rite the fo""o$in! sim"taneos "ineare%ations as a matrix e%ation.

+ x / 2y * 0 x / -y *

Hence, ca"c"ate the va"es of x and y sin! matrices.

&(a)

−

−

− 35

24

2

1(b) x * , y *-'

Answer :

- (a) 3tate matrix M if

23

15 M *

23

15

(b) 4sin! matrices, ca"c"ate the va"e of and $ that satisf# the fo""o$in! matrixe%ations:

−=

2

6

23

15

w

u

&(a)

10

01,(b) u * /2, w * -'

Answer :

2

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(a) iven that1 2 1 0

3 5 0 1 A

− = ÷ ÷−

find

matrix A.

(b) Hence, sin! the matrix method, find theva"e of r and s $hich satisf# thesim"taneos e%ations be"o$.

–r 5 2s * /- /+r 5 s * /

&(a)A*

−

−13

25 (b)r * /2, s * /+ '

Answer :

6 iven matrix P *4 5

6 8

÷

and matrix

PQ *1 0

0 1

÷

(a) Find the matrix Q.

(b) Hence, ca"c"ate b# sin! the matrixmethod, the va"es of m and n thatsatisf# the fo""o$in! sim"taneos"inear e%ations :

-m 5 n * 0 6m 5 7n * 18

&(a)9*

−

−

46

58

2

1 (b)m * +, n * /1'

Answer :

3

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0 t is !iven that matrix P *2 5

2k

÷−

does not

have an inverse matrix. (a) Find the va"e of k .

(b) f * 1, find the inverse matrix of ; andhence, sin! matrices, find the va"esof x and y that satisf# the fo""o$in!sim"taneos "inear e%ations.

2 x 5 y * 1+ x / 2y * /0

&(a) k * / -(b) x * /1, y * +'

Answer :

7(a) Find matrix M sch that

2 4

1 3

÷

M *2 4

1 3

÷

(b) 4sin! matrices, ca"c"ate the va"es of x and y that satisf# the fo""o$in! matrixe%ation.

2 4 6

1 3 5

x

y

= ÷ ÷ ÷

&(a)M*

10

01 x * /1, y * 2'

Answer :

4

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NAME : ________________________

Form : _______________Matrices_22

(a) Find the inverse of matrix3 1

5 2

− ÷−

(b) Hence, sin! matrices, ca"c"ate the va"esof d and e that satisf# thefo""o$in! sim"taneos e%ations :

2d / e * 0d / e * 16

&(a)

−−35

12(b) d * +, e * /1'

Answer :

18 iven matrix M *1 2

2 5

− ÷

, find

(a) the inverse matrix of M

(b) hence, sin! matrices, the va"es of u and v that satisf# the fo""o$in!sim"taneos e%ations :

u / 2v * 72u 5 v * 0

&(a)

− 12

25

9

1 (b)u * 6, v * /1'

Answer :

5

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(TRIAL SP !""6#

11.(a) iven that the inverse of

−

12

1

41

is

−−

−

12

1

41 n

m . Find the va"es of m and n.

(b) Hence, sin! matrices, ca"c"atethe va"es of x and # $hich satisf#the fo""o$in! sim"taneos "ineare%ations:

− x $ -y = %

2

1 x $ y = 6

& a) m *

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(TRIAL SP !""'#

1+.(a) iven that

=

−

−−

10

01

01

32

p1

30k ,

find the va"es of and . (b) Hence, sin! matrices, ca"c"ate

the va"es of x and # $hich

satisf# the fo""o$in! sim"taneos"inear e%ations:

32

23

−=−=

y x

y

& a) = *

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(TRIAL S.LA/G*R !""6#

1.(a) 9 is a 2 × 2 matrix $here

10

019 *

−1213

Find the matrix 9. (b) 4sin! matrices, ca"c"ate the va"es

of 0 and k $hich satisf# thefo""o$in! sim"taneos "inear e%ations:

142

63

=−=+

k h

k h

& (a)

1- 2

1 3 (b) h * -, *

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(TRIAL MRSM !""

1. (a) iven that matrix ; is 2 x 2 matrix

$here ; =

43

65

10

01.

Find the matrix ;.

(b) Hence, sin! matrices, ca"c"ate theva"es of m and n $hich satisf# thefo""o$in! sim"taneos "ineare%ations.

2m / +n * 7

/2

3 m 5

2

5n * 6

&(a)

2

5

2

3-

3- 2

(b) x * 06, # * -7 '

(TRIAL )*+*R !""

28. (a) iven that matrix A *

−

−

57

46,

find the inverse matrix of A.

(b) Hence, sin! matrices, ca"c"ate theva"es of x and # $hich satisf# thefo""o$in! sim"taneos "ineare%ations.

6m / -n * 7 /0m 5 n * /

& a)

3 2

7

2 2

5

b) m * 2, n * 1'

10

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21 (a) Find the inverse matrix of

31

42

(b) 4sin! matrices, ca"c"ate the va"e of k and of m that satisf# the fo""o$in!sim"taneos "inear e%ations:

2 5 -m * 18

5 +m * 7

&(a)

−

−

12

1

22

3

b) * /1, m * + '

22 (a) iven that matrix

−−

=23

48 P .

Find the matrix Q if

=

10

01 PQ .

(b) Hence, b# sin! matrices, ca"c"ate

the va"e of u and of w that satisf#the fo""o$in! matrix e%ation:

7 5 -$ * - /+ / 2$ * 8

&(a)

−−

24

3

12

1

(b) * 2, $ * /+'

11

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2+ iven that

1

131

r k is the

inverse matrix of

−32

11.

(i) Find the va"e of r and of k .(ii) Hence, b# sin! matrices, find the

va"e of x and of y $hich satisf# thefo""o$in! e%ation:

x / # * 112x 5 +# * 2

&(i)r * /2, * (ii)x * 0, # * /- '

2- iven matrix

−

−=

23

56 P , matrix

−

=63

51 k

mQ and matrix

=

10

01 PQ .

(a). Find the va"e of k and of m.(b). Hence, b# sin! matrices, find the va"e

of x and of y $hich satisf# the fo""o$in!matrix e%ation:

6x / # * -+x / 2# * 0

&(a). * /2, m * +.(b). x * , # * 18'

12

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NAME : ________________________

Form : ______________Matrices_2-

2 (a) iven the matrix e%ation

=

−

−

10

01

14

5

54

31 pk .

Find the va"e of and of k .(b) Hence, b# sin! matrices, find the

va"e of x and of y $hich satisf# thefo""o$in! e%ation:

x / +# * 7-x 5 # * 1

&(a) = 3 , k = 110.(b) x * , # * /1'

26 (a) iven

=

−

−

−

−

10

01

2

93

32

981

mn,

find the va"e of m and of n.

(b) 4sin! matrices, ca"c"ate the va"e of u and of w that satisf# the fo""o$in!matrix e%ation:

7 / $ * 2 / +$ * /1 &(a) m * 7 , n = –6 . (b) * -, $ * +.'

13

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20 M is a 22× matrix sch that

=

−

−

10

01

45

23 M

(a). Find matrix M .(b). rite the fo""o$in! sim"taneos "inear e%ation as matrix e%ation:

+x / 2# * 12

x / -# * 22Hence, sin! matrices, ca"c"ate the va"e of x and of y .

&(#

−

−=

2

3

2

5

12

M (b). x * 2,

# * /+'

27 (a) Find the matrix M if

=

23

15

23

15 M .

(b) ># sin! matrices, ca"c"atethe va"e of u and of w that satisf#the fo""o$in! matrix e%ation:

5 $ * /6

+ 5 2$ * 2

&(a)

=

10

01 M . (b) * /2, $ * -. '

14

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2 iven that matrices

−

−=

64

21 K ,

−

−=

m L

2

13

and

=

10

01 I ,

(i) Find the va"e of m if L =I (ii) ># sin! matrices, f ind the va"e of 4

and of u $hich satisf# the fo""o$in!e%ation:

t / 2 * 6-t / 6 * 22

&(i) m * ? (ii) t * -, * /1.'

+8 t is !iven that matrix

=

47

13 P and

matrix

=

10

01 PQ

(a) Find matrix Q.(b) Hence, b# sin! matrices, ca"c"ate

the va"e of x and of y that satisf# bothof the fo""o$in! e%ations:

+x 5 # * 0x 5 -# * 18

& (a).

−

−

=

5

3

5

7

5

1

5

4

Q (b) x * 2, # * /1'

15

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+1.Find the inverse matrix of *

−21

43

.

4sin! matrices, ca"c"ate the va"es of x and #$hich satisf# both of the fo""o$in! e%ations.

+x – -# * 0 x 5 2# * -

& (a)

− 31

42

10

1

(b) x * + , # * ? '

+2 a) Find the inverse matrix of

−21

43

.

b) 4sin! matrices, ca"c"ate the va"es of m andn that satisf# the fo""o$in! sim"taneos"inear e%ations.

+m 5 2n * -

-m 5 n * 10

. & a. 10

1

− 31

42

b. m * /2 , n * '

16

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NAME : ________________________

Form : _______________Matrices_2

++. F is a 2 2× matrix $here F

−−45

12

*

10

01

.

Find the matrix F. rite the fo""o$in! sim"taneos "inear

e%ations as a matrix e%ation.

2= / % * 18= / -% * +-

Hence, ca"c"ate the va"es of = and % sin!matrices.

&a. 3

1

−

−−

25

14

b. = *2, % * /6'

+-. iven that H *

k 2

35

.

@a"c"ate the va"e of for $hich matrix H hasno inverse matrix.

iven that * 2,find the inverse matrix of H. Hence, ca"c"ate the va"es of d and e $hich

satisf# the fo""o$in! matrix e%ation.

22

35

e

d

*

8

18

. & a. * 6 4

1

−

−52

32

b. d * +, e * 1'

17

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+．t is !iven that the matrix e%ation

q

1

− 2316

−−6

12

p*

10

01

. a) Find the va"e of = and of %. b) rite the fo""o$in! sim"taneos "inear

e%ations as a matrix e%ation.

6x 5 # * +x/2# */1+

Hence, se the matrix method to ca"c"ate theva"e of x and of #.

. & a. = * /+, % * /1 b. x * 1+ , # * 0'

+6. a) t is !iven that matrix A *

−n

m

4

1

and its

inverse matrix * 12

1

− m4

14

. Find the va"e of mand of n. b) 4se the matrix method to ca"c"ate the va"es of

x and of # $hich satisf# the fo""o$in! matrix e%ation.

−

=

19

11

y

x A

& a. m * 2, n * - b. x * / 212, # * -16 '

18

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+0. a. iven matrix ; *

−43

21 and matrix 9

sch that ;9 *

10

01.Find matrix 9.

b. t is !iven that

−−53

64

h g

f e*

10

01.

Find the va"es of e,5, and 0.

& a.9 *

−10

1

10

3

5

1

5

2

b. e *2

5 , f * /+, ! *

2

3 , h

* /2'

38. a) Find the inverse matrix of

−21

43.

b) 4sin! matrices, ca"c"ate the va"esof m and n that satisf# the fo""o$in!sim"taneos "inear e%ations.

+m 5 2n * - -m 5 n * 10

& a. 10

1

− 31

42 b. m * /2 , n * '

19

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39. F is a 2 2× matrix $here

F

−−45

12*

10

01.

a) Find the matrix F.b) rite the fo""o$in! sim"taneos

"inear e%ations as a matrixe%ation. 2 / 7 * 18

/ -7 * +- Hence, ca"c"ate the va"es of

and 7 sin! matrices.

&a.3

1

−

−−

25

14 b. = *2, % * / 6'

40. iven that H *

k 2

35.

a)@a"c"ate the va"e of for $hich matrix H hasno inverse matrix.

b)iven that * 2,find the inverse matrix of H. c)Hence, ca"c"ate the va"es of d and e $hich

satisf# the fo""o$in! matrix e%ation.

22

35

e

d *

8

18

&a. *6, b.4

1

−

−52

32 (c) d * + , e * 1 '

20

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NAME : ________________________

Form : _______________Matrices_26

41. t is !iven that the matrix e%ation

q

1

− 2316

−−6

12

p*

10

01.

a) Find the va"e of = and of %. b) rite the fo""o$in! sim"taneos "inear

e%ations as a matrix e%ation. 6x 5 # * +x/2# * /1+

Hence, se the matrix method to ca"c"ate theva"e of x and of #.

& a. = * /+, % * /1 b. x *3

1 , # * 0'

42. a) t is !iven that matrix A *

−n

m

4

1 and

its inverse matrix *12

1

− m4

14.

Find the va"e of m and of n.

b) 4se the matrix method to ca"c"ate theva"es of x and of # $hich satisf# thefo""o$in! matrix e%ation.

A

y

x *

12

12

&a. m * 2, n * - , b. x * , # * /2'

21

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43 M is a 2 x 2 matrix where M (45

23

−−

) = (10

01

)

a)i!" the matrix M

b)#rite the $%&&%wi!' sim&ta!e%s &i!ear

eati%! as a matrix eati%!.

3 x – 2 y = 7

5 x – 4 y = 9

*e!+e , +a&+&ate the -a&es %$ x a!" y si!'

matri+es.

& a. ( )35

24

2

1

−−

−= M b. x * , y * -'

44 a) he i!-erse matrix %$

−−65

43is m

−−

35

6 p

i!" the -a&e %$ m a!" %$ p.

/ m = , p = 4

b) si!' matri+es, +a&+&ate the -a&e %$ x a!"%$ y that satis$y the $%&&%wi!' sim&ta!e%s

&i!ear eati%!

3 x – 4 y = – 1

5 x – 6 y = 2 &a. m * ? , = * - b. x * 0 , y * 112'

22

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45 t is 'i-e! that matrix P =

−31

52 a!" matrix

Q = k

− 213 h

s+h that PQ =

10

01.

a)i!" the -a&e %$ k a!" %$ h.

b)si!' matri+es , $i!" the -a&e %$ x a!" %$ y that satis$y the $%&&%wi!' sim&ta!e%s &i!ear

eati%!

2 x – 5 y = –17

x 3 y = 8

&a. k * 111 , 0 * b. x * / 1 , y * +'

46 a)t is 'i-e! that

n

2

1

21

is the i!-erse matrix %$

−

−21

43. i!" the -a&e %$ n.

b)#rite the $%&&%wi!' sim&ta!e%s &i!eareati%!s are matrix eati%!

3u – 4v = –5

– u + 2v = 2

he!+e , si!' matri+es , +a&+&ate the -a&e %$

u a!" %$ v. &a. n * +2 (b) * / 1 , v * ? '

23

8/20/2019 2012 204 Matrices

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-0. M is a 2 × 2 matrix $hereM

−−45

23*

10

01.

(a) Find the matrix M .(b) rite the fo""o$in! sim"taneos "inear

e%ation as a matrix e%ation.+ x / 2y * 0

x / -y * Hence, ca"c"ate the va"es of x and #sin! matrices. &3;M Nov 288+'

(a)

−−

−35

24

2

1 (b) x * , # * - '

-7. (a) Find the inverse matrix of .65

32

(b) 4sin! matrices, ca"c"ate the va"es of xand y $hich satisf# the fo""o$in!sim"taneos "inear e%ations.

2 x 5 +y * 1 x 5 6y * /2

&3;M Bne 288-'

(a) (b) x * /-, y * +

24

−

−−

25

36

3

1

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NAME : ________________________

Form : _____________Matrices_20

-. (a) Che inverse matrix of

−

−

65

43 is

−

−

35

6 pm .

Find the va"es of m and (b) 4sin! matrices, ca"c"ate the va"es of x

and y $hich satisf# the fo""o$in!sim"taneos "inear e%ations.

+ x / -y * /1 x / 6y * 2 &3;M Nov. 288-'

(a) m * ? =* - (b) x * 0 # * 112

8. P is a 2 × 2 matrix $here

−43

21P *

10

01.

(a) Find the matrix P .

(b) rite the fo""o$in! sim"taneos "ineare%ation as a matrix e%ation.

x / 2y * 7+ x 5 -y *

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1. t is !iven that matrix P *

−31

52, matrix Q *

− 213 h

k sch that PQ *

10

01.

(a) Find the va"e of k and of 0.(b) 4sin! matrices, ca"c"ate the va"es of x

and y $hich satisf# the fo""o$in!sim"taneos "inear e%ations.

2 x / y *

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27/49

+. (a) t is !iven that

n2

1

21

is the inverse

matrix of

−

−21

43. Find the va"e of

n. (b) rite the fo""o$in! sim"taneos "inear

e%ation as matrix e%ation: +u / -v * /

/u 5 2v * 2 Hence, sin! matrices, ca"c"ate the

va"e of u and of v .

&3;M Nov. 2886' & (a) n * +2 (b) * / 1 v * ? '

-. (a) Find the inverse matrix of

−

43

21.

(b) rite the fo""o$in! sim"taneos "ineare%ation as matrix e%ation:

x / 2y * - + x 5 -y * 2

Hence, sin! matrices, ca"c"ate the va"eof x and of #.

&3;M Bne 2880'(a)(-# x * 2,y * /1

− 13

24

10

1

27

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Name : __________________________________

Form : _________________Matrices_27

0. Che inverse matrix of

74

32 is

−

2

371

mk .

(a) Find the va"e of m and of .

(b) rite the fo""o$in! sim"taneos "inear

e%ations as matrix e%ation:

574

132

=+−=+

y x

y x

Hence, sin! matrix method, ca"c"ate theva"e of x and of y .

& & (a) x * /11, y *0 (b) * 2 m * / - '

7. t is !iven that matrix ; *

−−47

58

(a) Find the inverse matrix of P (b) rite the fo""o$in! sim"taneos "ineare%ations as matrix e%ation Hence , b# sin! matrix method ,ca"c"ate the

va"e of x and the va"e of y

147

258

=−

=−

y x

y x

(a)

−

−

3

8

3

7

3

5

3

4

(b) x * /1. y * /2

29

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. M is a 2 × 2 matrix $hereM

−−45

23*

10

01.

(a) Find the matrix M .(b) rite the fo""o$in! sim"taneos

"inear e%ation as a matrix e%ation.+ x / 2y * 0

x / -y * Hence, ca"c"ate the va"es of x and# sin! matrices. &3;M Nov 288+'

(a)

−

−−=

35

24

2

1 M (b) x * y * -

68. (a) Find the inverse matrix of .65

32

(b) 4sin! matrices, ca"c"ate the va"es of x and y $hich satisf# the fo""o$in!sim"taneos "inear e%ations.

2 x 5 +y * 1 x 5 6y * /2

&3;M Bne 288-'

(a)

−

−−

25

36

3

1 (b) x * /-, # * +

30

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61. (a) Che inverse matrix of

−−65

43 is

−−

35

6 pm .

Find the va"es of m and (b) 4sin! matrices, ca"c"ate the va"es of

x and y $hich satisf# the fo""o$in!

sim"taneos "inear e%ations.+ x / -y * /1 x / 6y * 2

&3;M Nov. 288-'

(# m *2

1, * - (b) x * 0 , y *

2

11

62. P is a 2 × 2 matrix $here

−−45

23P *

10

01.

(a) Find the matrix P .

(b) rite the fo""o$in! sim"taneos

"inear e%ation as a matrix e%ation. x / 2y * 7+ x 5 -y *

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6+. t is !iven that matrix P *

−31

52,

matrix Q *

− 213 h

k sch that PQ *

10

01.

(a) Find the va"e of k and of 0.(b) 4sin! matrices, ca"c"ate the va"es of

x and y $hich satisf# the fo""o$in!sim"taneos "inear e%ations.

2 x / y *

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Name : __________________________________

Form : __________________Matirces_2

6. (a) t is !iven that

n2

1

21

is the inverse

matrix of

−

−21

43. Find the va"e of n.

(b) rite the fo""o$in! sim"taneos"inear e%ation as matrix e%ation:

+u / -v * / /u 5 2v * 2 Hence, sin! matrices, ca"c"ate

the va"e of u and of v .

&3;M Nov. 2886'

(# n = 38! (-# u * /1, v *12

66. (a) Find the inverse matrix of

−

43

21.

(b) rite the fo""o$in! sim"taneos"inear e%ation as matrix e%ation:

x / 2y * - + x 5 -y * 2

Hence, sin! matrices, ca"c"ate theva"e of x and of #.

&3;M Bne 2880'

(a)

− 13

24

10

1 (b) x * 2 # * / 1

33

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60. (a) iven thatm

1

−

−35

24

−

−45

2n*

10

01,

find the va"e of m and of n.(b) 4sin! matrices, ca"c"ate the va"e of x

and of # that satisf# the fo""o$in! matrixe%ation:

−

−

35

24

y

x *

2

1

&3;M Nov 2880'

(a) m * /2 , n * + (b) x *12 , # * +2

67. (a) iven that

p

1

−

1

24

q

− 43

21*

10

01.

Find the va"e of and of 7.(b) rite the fo""o$in! sim"taneos

"inear e%ations as matrix e%ation:

1143

22

−=+−=+ y x

y x


& 3;M Bne 2887 '

(# 7 * +, * 18 (b) x * +, y * / ?

34

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6. Che inverse matrix of

74

32 is

−−2

371

mk

.(a) Find the va"e of m and of .

(b) rite the fo""o$in! sim"taneos "inear

e%ations as matrix e%ation:

574

132

=+−=+

y x

y x


&3;M Nov 2887'

(a) * 2 , m * / - (b) x * /11, y *0

08. t is !iven that matrix ; *

−−47

58

(a) Find the inverse matrix of P (b) rite the fo""o$in! sim"taneos "inear

e%ations as matrix e%ation Hence , b# sin! matrix method ,ca"c"atethe va"e of x and the va"e of y

147

258

=−

=−

y x

y x

3;M Bne 288

(a)

−

−

3

8

3

7

3

5

3

4

(b) x * /1. y * /2

35

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01. t is 'i-e! that matrix is .54

32

−

−= A

Diberi bahawa marik! .54

32

−−

= A

(a) i!" the i!-erse matrix %$ A.

"ari marik! !#ng!ang bagi A.

(b) #rite the $%&&%wi!' sim&ta!e%s &i!ear

eati%!s as matrix eati%!

$u%i! per!amaan %inear !erenak beriku da%am

benuk per!amaan marik!&

1354

732

=−=−

y x

y x

*e!+e, si!' matrix meth%", +a&+&ate the

-a&e %$ x a!" %$ y.

'eeru!nya( menggunakan kaedah marik!(

hiung ni%ai x dan ni%ai y.

& 3;M ND 288 '(

a)

−

−

12

2

3

2

5

(b) x * 2 # * / 1

02, Che inverse matrix of is &0M'

Matris sonsan! ba!i ia"ah

a)Find the va"e of m and

@ari ni"ai m dan b)rite the fo""o$in! sim"taneos "ineare%ations as matrix e%ation.C"is =ersamaan "inear serenta berit da"ambent =ersamaan matris.

Hence , sin! the matrix method , ca"c"atethe va"e of x and #.3etersn#a , den!an men!!naan aedahmatris, hitn! ni"ai xand ni"ai #.

3;M ND 2818

(a) m * - , * 122 (b) x * +2 # * / 1'

36

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Name : _______________________________________________

Form : _________________________ atri+es210

0+) a) iven that E is the matrix

34

45, find the

matrix F sch that EF *

10

01

b) 4sin! matrices, ca"c"ate the va"es ofm and n $hich satisf# the fo""o$in! matrixe%ation.

34

45

n

m*

− 2

6

(a)

−

−=

54

43 ) inver!e (b) m * / 26 , n * +-

0-) hen so"vin! the matrix e%ation

− 3203

y

x*

11

12, it is fond that

y x *

k 1

!r q p

1112

a) Find the va"es of , =, %, r and s.b) Hence, find the va"es of x and #

(a) * / , = * /+, % * 8 , r * / 2 , s * + (b) x * - # * / 1

37

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0) a) Find the inverse matrix of *

−21

43

b) 4sin! matrices, ca"c"ate the va"es of xand # $hich satisf# both of the fo""o$in!e%ations.

+x / -# * 0 x 5 2# * -

(a)

−

=31

42

10

1G Inver!e (b) x + , # * ?

06) a) iven that the inverse matrix of

23

54

is

−7

4

7

5

k

h

, state the va"e of h and .

b) 4sin! matrices, ca"c"ate the va"es of xand # $hich satisf# both of the fo""o$in!e%ations.

-x 5 # * /1 +x 5 2# * 7

(a) h * / 20 , * +0 (b) x * 6 # * /

38

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00) a) iven that F is the matrix

n

m

10

1 and

the inverse matrix of F is10

1

−

−510

1n

find the va"e of m and n. b) Hence, ca"c"ate the va"es of x and #

$hich satisf# the fo""o$in! matrixe%ation.

F

y

x*

6

4

(a) ) m * , n * - (b) x * 1 , # * /1

07) a) Find the inverse matrix of

−−86

11

b) Hence, ca"c"ate the va"es of h and $hich satisf# the fo""o$in! matrixe%ation.

−−86

11

k

h*

−

15

2

(a)

−−

=16

18

2

1 ) inver!e (b) h * 12 , *

+2

39

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0) a) iven that M is the matrix

− 13

11

and the inverse matrix of M isk

1

1

1

n

m,

Find the va"es of , m and n b) hence, ca"c"ate the va"es of x and #

$hich satisf# the fo""o$in! matrixe%ation.

M

y

x*

− 5

3

(a) * - , m * /1, n * + (b ) x * 2 # * 1

78) iven that E *

23

45 and F *

−

−10

84

k ,

a) Find the va"e of $here EF *

40

04

b) 3tate the inverse matrix of E c) Hence, sin! matrices, find the va"e of x

and # $hich satisf# the fo""o$in! matrixe%ations.

23

45

y

x*

4

6

(a) * 6 (b)

−

−−=

53

42

2

1 * Inver!e (c) x*2 ,#

* / 1

40

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Name : _________________________________

Form : _________________Matices 2_11

71) a) Che inverse matrix of

− 4312

is

m

−−

2

14

p, find the va"e of m and =.

b) 4sin! matrices, ca"c"ate the va"e of xand # $hich satisf# the fo""o$in!sim"taneos "inear e%ations.

2x 5 # * - +x / -# * 10 (a) m * / 111 , = * / + (b) x * + # * / 2

72) a) t is !iven that matrix M *

21

43

,

find the inverse matrix of M b) 4sin! matrices, find the va"es of x and #

$hich satisf# the fo""o$in! e%ations.

+x 5 -# * /11 x 5 2# * /0

(a)

−

−=

31

42

2

1 M Inver!e (b) x* + , # * /

41

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7+) iven that the inverse matrix of

−− 32

12

is

−−2

131

nm.

(a) Find the va"e of m and of n.(b) Hence, sin! matrices, ca"c"ate the

va"es of d and e in the fo""o$in!sim"taneos "inear e%ations : 2d $ e * 0

/2d / +e * /1

(a) m * / - , n * 2 (b) d * e * , # * / + '

7-) (a)iven

=

−

−

10

01

12

3

42

311 n

m,

find the va"e of m and n.

(b) 4sin! matrices, ca"c"ate the va"e of x and y that satisf# the fo""o$in! matrix

e%ation :

=

−6

8

42

31

y

x

(a) m * 18 , n * - (b) x * # * / 1

42

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7) (a) i-e! that matrix

−−

=43

21 P

a!" matrix

−=

2

1

12

!Q .

Diberi marik!

−−

=43

21 P dan

marik!

−=

2

1

12

!Q .

i!" the -a&e %$ ! s+h that

=

10

01 PQ .

"ari ni%ai ! dengan keadaan 1 0

0 1 PQ

= ÷

.

(b) #rite the $%&&%wi!' sim&ta!e%s &i!eareati%!s as a matrix eati%!.

$u%i! per!amaan %inear !erenak berikuda%am benuk per!amaan marik!.

1143

42

=+−=−−

y x

y x

*e!+e, si!' matri+es, +a&+&ate the

-a&e %$ x a!" %$ y. 'eeru!nya( dengan menggunakan kaedah

marik!( hiung ni%ai x dan y . (a) s * +2 (b) x * + , # * ?

43

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86. (a)he i!-erse matriks %$

−−

41

52 is

p

1

−−21

5r

i!" the -a&e %$ p a!" %$ r

Marik! !#ng!ang bagi

−−

41

52 ia%ah

p

1

−

−

21

5r

"arikan ni%ai p dan ni%ai r

(b) si!' matri+es, +a&+&ate the -a&e %$ x a!" %$ y

that satis$y the $%&&%wi!' sim&ta!e%s &i!er

eati%!

Dengan menggunakan kaedah marik!( hiungkan

ni%ai x dan ni%ai y yang memua!kan

pe!amaan %inear !erenak beriku &

2 x 5 y = 12

x 4 y = 3 & (a) = * / + r * - (b) x * / 11 , # * / 2 '

87. (a) i-e!

=

−

−

10

01

6

12

26

14

pk ,

$i!" the -a&e %$ k a!" %$ p.

Diberi

=

−

−

10

01

6

12

26

14

pk (

enukan ni%ai k dan ni%ai p.

(b) *e!+e, by si!' matri+es, +a&+&ate the -a&e %$ xa!" %$ y that satis$y the $%&&%wi!' sim&ta!e%s

&i!ear eati%!s.

'eeru!nya( dengan menggunakan kaedah

marik!( hiungkan ni%ai x dan ni%ai y yang

memua!kan per!amaan %inear !erenak beriku

2226

134

=+−−=−

y x

y x

& (a) * ? = * - (b) x * / 2 # * '

44

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88. (a) t 'i-e! that matrix P =

−23

1n, $i!" the

-a&e %$ n i$ matrix P !%t ha-e i!-erse matrix.

$ n 5 ($i!" the i!-erse matrix %$ P .

Diberi marik! P

−23

1n ( ,ari ni%ai n -ika

marik! P idak mempunyai !#ng!ang . ika n 5 ( ,ari marik! !#ng!ang bagi P.

(b) *e!+e, by si!' matri+es, +a&+&ate the -a&e %$ x

a!" %$ y that satis$y the eati%!s.

'eeru!nya( dengan menggunakan kaedah marik!(

hiungkan ni%ai v dan ni%ai w yang memua!kan

pe!amaan iu.

−

=

−

1

7

23

15

y

x

& (a) n * / +2 inverse ; *

− 53

12

13

1 x * 1 , # * / 2 '

:ame

%rm atir+es 212

89. M is a matrix 2 × 2 s+h that M 4 3 1 0

7 5 0 1

= ÷ ÷−

M ia%ah !au marik! 2 × 2 dengan keadaan M

4 3 1 0

7 5 0 1 = ÷ ÷−

(a) i!" the matrix M.

"arikan marik! M.


eati%!s as a matrix eati%! .

$u%i!kan per!amaan !erenak beriku da%am

benuk per!amaan marik! .

4 x 3 y = 23

7 x 5 y = 11

*e!+e, by si!' matri+es, +a&+&ate the -a&e

%$ x a!" %$ y that satis$y the eati%!s.



memua!kan pe!amaan iu.

/ (a) !-erse =

−

−−−47

35

41

1 (b) x * 2 ,

# * '

45

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90. i-e! that matrix A =

−−

48

23 a!"

matrix / =

−

h

k

8

24 s+h that A/ =

10

01.

Diberi bahawa marik! A

−−

48

23 dan

marik! /

−h

k 8

24 dengan kea"aa!

A/ =

10

01.

(a) i!" the -a&e %$ k a!" -a&e %$ h.

"arikan ni%ai k dan ni%ai h.

(b) si!' matri+es, $i!" the -a&e %$ x a!"

%$ a!" %$ y that satis$y the $%&&%wi!'

sim&ta!e%s &i!ear eati%!s.

Dengan menggunakan kaedah marik!(

hiungkan ni%ai x dan ni%ai y yang

memua!kan per!amaan %inear !erenak

beriku &

3 x 2 y = 6

8 x 4 y = 4

& (a) * h * / + (b) x * - , # * '

91. i-e! that matri+es P =2

2 3

k ÷−

a!"

Q =2 1

3 2

÷−

,

Diberi bahawa marik! P 2

2 3

k

÷− dan

Q 2 1

3 2

÷−

(

(a) i!"

"arikan

(i) the -a&e %$ k , i$ matrix P has

!% i!-erse

ni%ai k( -ika marik! P idak

mempunyai !#ng!ang(

(ii) the i!-erse %$ matrix Q.

marik! !#ng!ang bagi Q.

(b) si!' matri+es, +a&+&ate the -a&e %$ x a!" the

-a&e %$ y that satis$y the $%&&%wi!'

sim&ta!e%s eati%!s

Dengan menggunakan kaedah marik!(

hiungkan ni%ai x dan ni%ai y yang

memua!kan per!amaan %inear !erenak

beriku &

2 x y = 4

3 x ; 2 y = 13 & (a)(i) * (ii) h * / + (b) x * + , # * / 2 '

46

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92. i-e! M is a 2 × 2 matrix s+h that Diberi M ia%ah !au marik 2 × 2 dengan

keadaan

M

41

63=

10

01

0a1 i!" the matrix M.

"arikan marik! M.

0b1 #rite the $%&&%wi!' sim&ta!e%s &i!ear




3 x 6 y = 12

x 4 y = 10

*e!+e, by si!' matri+es, +a&+&ate the

-a&e %$ x a!" %$ y that satis$y the eati%!s.




& (a) M *

−

−31

64

6

1 (b) x * / 2 # * + '

93. (a) i-e! that x

−−

67

5 y

−−57

46=

10

01,

$i!" the -a&e %$ x a!" %$ y.

Diberi x

−−

675 y

−−5746 =

1001 ,

,arikan ni%ai x dan ni%ai y.





6v – 4w = 4

7v – 5w = 7

*e!+e %r %therwise $i!" the -a&es %$ v a!" w .

& (a) x * / ? # * - (b)

=

−−

7

4

57

46

w

v

v * / - , $ * / 0 '

47

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94. i-e! M is a 2 × 2 matrix s+h that Diberi M ia%ah !au marik 2 × 2 dengankeadaan

M

41

63=

10

01

0a1 i!" the matrix M.

"arikan marik! M.

0b1 #rite the $%&&%wi!' sim&ta!e%s &i!ear




3 x 6 y = 12

x 4 y = 10

*e!+e, by si!' matri+es, +a&+&ate the -a&e

%$ x a!" %$ y that satis$y the eati%!s.




& (a)

−

−31

64

6

1 (b)

=

10

12

41

63

y

x x * /

2 , # * +'

95. i-e! A =

n

m

1

5 a!" the i!-erse matrix %$ A is

2

1

−

−51

8n

Diberi A

n

m

1

5 dan marik! !#ng!ang bagi A

ia%ah2

1

−

−51

8n

(a) i!" the -a&e %$ m a!" n.

0a1 "arikan ni%ai bagi m dan n.


eati%!s as matrix eati%!

0b1 $u%i! per!amaan %inear !erenak beriku

da%am benuk per!amaan marik!&

5 x 5 8 y = 4

x 5 2 y = /2

*e!+e, si!' matri+es, +a&+&ate the -a&e %$ x a!"

%$ y.

'eeru!nya( dengan menggunakan kaedah marik!(

hiungkan ni%ai x dan ni%ai y

& (a) m * 7 n * 2 (b)

−

=

2

4

21

85

y

x

x * 12 , # * / 0 '

48

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96. i-e! that matri+es P =2

2 3

k ÷−

a!" Q =2 1

3 2

÷−

,

Diberi bahawa marik! P 2

2 3

k ÷−

dan Q 2 1

3 2

÷−

(

(a) i!"

"arikan

(

Documents

2012 204 Matrices