chapter 13 ii matrices enrich
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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.
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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
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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
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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
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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 )
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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
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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 )
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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
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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
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