chapter 13 ii matrices enrich

8/3/2019 Chapter 13 II Matrices ENRICH

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13.1 Identity Matrix

a) If AI = IA = A, whereA is any square matrix, thenIis an identity orunit matrix.

b) I =

10

01is a 2 x 2 identity matrix.

13.2 Inverse Matrix

a) If A =

dc

ba, then inverse of A , A 1 =

bcad 1

ac

bd

b) When ad bc 0, the expression ad bc is the determinant of matrix A.

c) A 1 does not exist if the determinant is zero.

13.3 Matrix Equations

Important concept.

the two linear simultaneous

ax + by = p

cx + dy = q

can be changed into matrix form:

y

x

dc

ba=

q

p

13.3.1 Write the simultaneous Linear Equations as a Matrix Equations

Example 1 :

2x + y = 2

7x + 3y = 6

37

12

y

x=

6

2

Example 2 :

2x y = 2

7x + 3y = 6

37

12

y

x=

6

2

Example 3 :

2x + y = 2

-7x + 3y = 6

37

12

y

x=

6

2

Example 4 :

2x + y = 2

7x 3y = 6

3712

y

x=

6

2

Questions based on Examination Format

Matrices 1

CHAPTER 13

MATRICES


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Example : i) Find the inverse matrix of

85

21

ii) Using matrices, calculate the value of k dan m which satisfy the

following matrix equation :

85

21

m

k=

11

1

Solution :

i)

1

85

21

=)5)(2()8(1

1

15

28

=108

1

+

15

28

=2

1

15

28

ii)

85

21

m

k=

11

1

m

k=

1

85

21

11

1

m

k=2

1

15

28

11

1

m

k=2

1

++

)11(1)1)(5(

)11(2)1)(8(

mk =

21

+ + 115

228 =21

614 =

2

62

14

m

k=

3

7, hence k = 7 and m = 3

1) a) Given that E is the matrix

34

45, find the matrix F such that EF =

10

01

b) Using matrices, calculate the values of m and n which satisfy the following matrix

equation.

Matrices 2


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34

45

n

m=

26

2) When solving the matrix equation

3203

y

x=

11

12, it is found that

y

x=k

1

sr

qp

11

12

a) Find the values of k, p, q, r and s.

b) Hence, find the values of x and y

3) a) Find the inverse matrix of G =

21

43

b) Using matrices, calculate the values of x and y which satisfy both of the following

equations.

3x 4y = 7

x + 2y = 4

4) a) Given that the inverse matrix of

2354 is

7

4

7

5

k

h

, state the value of h and k.

b) Using matrices, calculate the values of x and y which satisfy both of the following

equations.

4x + 5y = -1

3x + 2y = 8

5) a) Given that F is the matrix

n

m

10

1and the inverse matrix of F is

10

1

510

1n,

find the value of m and n.

b) Hence, calculate the values of x and y which satisfy the following matrix equation.

F

y

x=

6

4

Matrices 3


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6) a) Find the inverse matrix of

86

11

b) Hence, calculate the values of h and k which satisfy the following matrix equation.

86

11

k

h=

15

2

7) a) Given that M is the matrix

13

11and the inverse matrix of M is

k

1

1

1

n

m,

Find the values of k, m and n

b) hence, calculate the values of x and y which satisfy the following matrix equation.

M

y

x=

53

8) Given that E =

23

45and F =

10

84

k,

a) Find the value of k where EF =

40

04

b) State the inverse matrix of E

c) Hence, using matrices, find the value of x and y which satisfy the following matrix

equations.

23

45

y

x=

4

6

9) a) The inverse matrix of

4312

is m

2

14

p, find the value of m and p.

b) Using matrices, calculate the value of x and y which satisfy the following simultaneous

linear equations.

2x + y = 4

3x 4y = 17

Matrices 4


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10) a) It is given that matrix M =

21

43, find the inverse matrix of M

b) Using matrices, find the values of x and y which satisfy the following equations.

3x + 4y = -11x + 2y = -7

PAST YEAR SPM QUESTIONS

November 2003

1. M is a matrix where M

45

23=

10

01

a) Find the matrix M

b) Write the following simultaneous linear equations as a matrix equation.

3x 2y = 7

5x 4y = 9

Hence, calculate the values of x and y using matrices.

( 6 marks )

July 2004

2. a) Find the inverse matrix of

65

32

.

b) Using matrices, calculate the value of x and of y thay satisfy the following

simultaneous linear equations:

265

132

=+

=+

yx

yx

( 6 marks )

November 2004

Matrices 5


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3. a) The inverse matrix of

65

43is m

35

6 p, find the value of m and ofp.

b) Using matrices, calculate the value of x and of y that satisfy the following

simultaneous linear equations.

3x 4y = -1

5x 6y = 2

( 6 marks )

July 2005

4. P is a 2 x 2 matrix such that

=

10

01

43

21P

a) Find the matrix P

b) Write the following simultaneous linear equations as a matrix equation:

643

82

=+

=

yx

yx

Hence, using matrices, calculate the value ofx and ofy.

( 6 marks )

November 2005

5. It is given that matrix P =

31

52and matrix Q = k

213 h

such that

PQ =

10

01

a) Find the value of k and ofh

b) Using matrices, find the value of x and of y that satisfy the following simultaneouslinear equations :

2x 5y = -17x + 3y = 8

( 6 marks )

July 2006

6. It is given that matrix M =

21

43.

(a) Find the inverse matrix of M(b) Write the following simultaneous linear equations as matrix equation:

6y2x

13y4x3

=+=

Hence, using matrices, calculate the value ofx and ofy.( 6 marks )

Matrices 6


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November 2006

7. (a) It is given that

n

2

1

21

is inverse matrix of

21

43. Find the value ofn.

(b) Write the following simultaneous linear equations as matrix equation:

22

543

=+

=

vu

vu

Hence, using matrices, calculate the value ofu and ofv( 6 marks )

Jun 2007

8. a) Find the inverse matrix of

43

21.

b) Write the following simultaneous linear equations as matrix equation:

243

42

=+

=

yx

yx

Hence, using matrix method, calculate the value ofx and ofy.( 6 marks )

November 2007

9. (a) Givenm

1

35

24

=

10

01

45

2n. Find the values of m and n.

(b) Using matrices, calculate the value of x and y that satisfy the following matrix

equation.

35

24

y

x=

2

1

( 7 marks )

Jun 2008

Matrices 7


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10. (a) Given thatp

1

1

24

q

=

10

01

43

21. Find the value ofp and q.

(b) Write the following simultaneous linear equations as matrix equation:

1143

22

=+=+

yx

yx

( 7 marks )

November 2008

11. The inverse matrix of 2 3

4 7

is

2

371

mk

(c) Find the value ofm and ofk.(d) Write the following simultaneous linear equations as matrix equation:

574

132

=+=+

yx

yx

Hence, using matrix method, calculate the value ofx and ofy.

( 6 marks )

Matrices 8


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ANSWERS

QUESTIONS ACCORDING TO EXAMINATION FORMAT.

1) a)

54

436) a)

2

1

1618

b) m = -26, n = 24 b) h =2

1, k =

2

3

2) a) k = -9, p = -3, q = 0, r = -2, s = 3 7) a) k = 4, m = -1, n = 3

b) x = 4, y = -1 b) x = 2, y = 1

3) a)10

1

31

428) a) k = 6

b) x = 3, y =2

1b)

2

1

53

42

c) x = 2, y = -14) a) h =

7

2, k =

7

39) a) m =

11

1

, p = -3

b) x = 6, y = -5 b) x = 3, y = -2

5) a) m = 5, n = 4 10) a)

2

3

2

1

21

b) x = 1, y = -1 b) x = 3 , y = -5

PAST YEAR SPM QUESTIONS

1. a)2

1

35

24b)

=

9

7

45

23

y

x, x = 5 , y = 4

2. a)

25

36

3

1b) x = -4, y = 3

3. a)

35

46

2

1b) x = 5, y =

2

11

4. a)

13

24

10

1b)

=

6

8

43

21

y

x, x = 2, y = -3

5. a) k =11

1

, h = 5 b) x = -1, y = 3

6. a)

31

42

10

1b)

=

2

5

21

43

v

u, x = 3, y =

2

3

7. a) n =2

3b)

=

2

5

21

43

v

u, u = -1, v =

2

1

Matrices 9


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8. a)4 21

3 110

b) x = 2, y = -1

9. a) m = 2, n = 3 b) x =1

2, y =

2

3

10. a) p = 10, q = 3 b) x = 3, y =1

2

11. a) k = 2, m = - 4 b) x = -11, y = 7

Matrices 10

chapter 13 ii matrices enrich

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