Matriks (matematik Tingkatan 5) 5z1r2
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NAME: FORM:
78) a)
TOPIC: MATRICES
b)
75)
3 4 1 2 a) Find the inverse matrix of G =
Hence, calculate the values of h and k which satisfy the following matrix equation.
1 1 6 8
b) Using matrices, calculate the values of x and y which satisfy both of the following equations. 3x – 4y = 7 x + 2y = 4 1 2 4 Inverse G 10 1 3 (a)
1 1 6 8 Find the inverse matrix of
inverse F (a)
1 1 8 2 6 1
h 2 k = 15 (b) h = 1/2 , k = 3/2
(b) x 3 , y = ½
79)
76) a)
h
k
4 5 3 2 Given that the inverse matrix of 5 7 4 7 , 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 (a) h = – 2/7 , k = 3/7 (b) x = 6
y=–5
77)
m 1 10 n and a) Given that F is the matrix 1 n 1 the inverse matrix of F is 10 10 5 find the value of m and n. b)
Hence, calculate the values of x and y which satisfy the following matrix equation.
x 4 y 6 F = (a) ) m = 5 , n = 4 (b) x = 1 , y = –1
is
1 1 3 1 a) Given that M is the matrix 1 1 m n 1 , and the inverse matrix of M is k Find the values of k, m and n b)
hence, calculate the values of x and y which satisfy the following matrix equation.
x 3 y 5 M = (a) k = 4 , m = –1, n = 3 (b )
x=2
y= 1
Inverse M (a)
80)
5 4 3 2 and F = Given that E =
a) b) c)
4 8 k 10 ,
4 0 0 4 Find the value of k where EF = State the inverse matrix of E Hence, using matrices, find the value of x and y which satisfy the following matrix equations.
5 4 3 2
x 6 y = 4
1 2 4 Inverse E 2 3 5 (c) x=2 ,y = – 1 (a) k = 6 (b)
81) a)
b)
2 1 3 4 is The inverse matrix of 4 1 p 2 , find the value of m and p. m Using matrices, calculate the value of x and y which satisfy the following simultaneous linear equations. 2x + y = 4 3x – 4y = 17 (a) m = – 1/11 , p = – 3
(b) x = 3
y= –2
82)
3 4 1 2 a) It is given that matrix M =
,
find the inverse matrix of M b) Using matrices, find the values of x and y which satisfy the following equations. 3x + 4y = –11 x + 2y = –7
1 2 4 2 1 3
(b) x= 3 , y = – 5
1 2 2 3 83) Given that the inverse matrix of 1 3 1 n 2 m . is (a) (b)
Find the value of m and of n. Hence, using matrices, calculate the values of d and e in the following simultaneous linear equations : 2d + e = 7 –2d – 3e = –1 (a) m = – 4 , n = 2
(b) d =
(a) Given that matrix
85)
2
and matrix
Diberi matriks
1 3 2 4
(a) m = 10 , n = 4
x 8 y 6
(b) x = 5 y = – 1
1 2 .
s
1 3
2 4
P
dan 2
Q
matriks
s
1
1 2 .
Find the value of s such that
1 0 PQ 0 1
.
Cari nilai s dengan keadaan 1 0
PQ (b) Using matrices, calculate the value of x and y that satisfy the following matrix equation :
2 4
1
Q
e=5,y=–3]
1 1 3 n 3 1 0 2 4 2 1 0 1 , 84) (a)Given m find the value of m and n.
1 3
P
0 1
.
(b) Write the following simultaneous linear equations as a matrix equation. Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks.
x 2y 4 3 x 4 y 11 Hence, using matrices, calculate the value of x and of y.
Seterusnya, dengan menggunakan kaedah matriks, hitung nilai x dan y . (a) s = 3/2
(b) x = 3 , y = ½
2 5
1 4 86. (a)The inverse matriks of is r 5 1 p 1 2 Find the value of p and of r 2 5 1 4 Matriks songsang bagi 1 r 5 ialah p 1 2 Carikan nilai p dan nilai r (b)
Using matrices, calculate the value of x and of y that satisfy the following simultaneous liner equation : Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan linear serentak berikut : −2x + 5y = 12 −x + 4y = 3 [ (a) p = – 3 r = 4 (b) x = – 11 , y
=–2]
87. (a) Given
4 6
k
1 2
2 6
1 1 0 p 0 1 , find the value of k and of p.
Diberi
4
k
6
1 2
2 6
1
1 p 0
, tentukan nilai k dan nilai p.
0
(b) Hence, by using matrices, calculate
the value of x and of y that satisfy the following simultaneous linear equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut
4 x y 13 6 x 2 y 22 [ (a) k = ½ p = 4 (b) x = – 2 y = 5 ]
1
88. (a) It given that matrix P = n 1 3 2 , find the value of n if matrix P not have inverse matrix. If n = 5,find the inverse matrix of P.
n 1 3 2 , cari Diberi matriks P = nilai n jika matriks P tidak mempunyai songsang . Jika n = 5, cari matriks songsang bagi P. (b) Hence, by using matrices, calculate the value of x and of y that satisfy the equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai v dan nilai w yang memuaskan pesamaan itu.
5 1 x 7 3 2 y 1 1 2 1 13 3 5 [ (a) n = – 3/2 inverse P =
Name : ___________________________ ____ Form : ___________________Matirces 2_12 89. M is a matrix 2 2 such that M 4 3 1 0 7 5 0 1
M ialah satu matriks 2 2 dengan
4 3 7 5 keadaan M
1 0
0 1
(a)
Find the matrix M. Carikan matriks M. (b) Write the following simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut
dalam
bentuk persamaan matriks . 4x + 3y = 23 7x − 5y = −11
x=1,y=–2]
Hence, by using matrices, calculate the value of x and of y that satisfy the equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan itu.
[ (a) Inverse M = 1 5 3 41 7 4
(b) x = 2 , y = 5 ]
3 2 8 4 and 90. Given that matrix A = 4 2 k 8 h such that AB matrix B = 1 0 0 1 . = 3 2 8 4 Diberi bahawa matriks A = dan
4 2 8 h dengan matriks B = k
keadaan
1 0 0 1 . AB = (a)
Find the value of k and value of h. Carikan nilai k dan nilai h.
(b)
Using matrices, find the value of x and of and of y that satisfy the following simultaneous linear equations. Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : − 3x + 2y = 6 − 8x + 4y = 4
[ (a) k = ¼ h = – 3
(b) x = 4 , y = 9 ]91.
2 k 2 3 and Given that matrices P = 2 1 3 2 , Q= 2 k 2 3 Diberi bahawa matriks P = dan
2 1 3 2 , Q= (a)
Find Carikan (i) the value of k, if matrix P has no inverse nilai k, jika matriks P tidak mempunyai songsang, (ii) the inverse of matrix Q. matriks songsang bagi Q.
(b) Using matrices, calculate the value of x and the value of y that satisfy the following simultaneous equations : Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : 2x + y = 4 3x ─ 2y = 13
[ (a)(i) k = ¼ (ii) h = – 3 (b) x = 3 , y = – 2 ]92.
1 4 6 6 1 3 (b) x = – 2 y = 3 ] [ (a) M =
Given M is a 2 2 matrix such that Diberi M ialah satu matrik 2 2 dengan keadaan
3 6 1 4 M
Write
the
following
simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut
find the value of x and of y.
Diberi
Carikan matriks M.
dalam
bentuk persamaan matriks . 3x + 6y = 12 x + 4y = 10
value of x and of y that satisfy the equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan itu.
5 y 7 6 x
6 4 7 5 =
1 0 0 1 , carikan nilai x dan nilai y. (b) Write the following simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut
dalam
bentuk persamaan matriks . 6v – 4w = 4
Hence, by using matrices, calculate the
6 4 7 5 =
1 0 0 1 ,
1 0 0 1 =
(a) Find the matrix M. (b)
93. (a)
5 y 7 6 Given that x
7v – 5w = 7 Hence or otherwise find the values of v and w . 6 4 v 4 7 5 w 7 [ (a) x = – ½ y = 4 (b) v=–4,w=–7]
94. Given M is a 2 2 matrix such that Diberi M ialah satu matrik 2 2 dengan keadaan
3 6 1 4 M
1 0 0 1 =
(a) Find the matrix M.
5 m
1 n 95. Given A = and the inverse n 8 1 1 5 matrix of A is 2 5 m
1 n Diberi A =
Carikan matriks M.
1
(b) Write the following
ialah 2
simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut dalam bentuk persamaan matriks . 3x + 6y = 12 x + 4y = 10
(a) (a)
8
1
5
n
Find the value of m and n. Carikan nilai bagi m dan n.
(b) Write the following simultaneous linear equations as matrix equation: (b) Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks: 5x + 8y = 4
Hence, by using matrices, calculate
the
x + 2y = –2
value
of x and of y that satisfy the equations.
[ (a) , y = 3]
dan matriks
songsang bagi A
Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan itu. 3 6 x 12 1 4 6 6 1 3 (b) 1 4 y 10 x = – 2
Hence, using matrices, calculate the value of x and of y. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y 5 8 x 4 1 2 y 2 [ (a) m = 8 n = 2 (b) x = 12 , y = – 7 ]
2 k 96. Given that matrices P = 2 3 and 2 1 3 2 , Q= 2 k 2 3 dan Diberi bahawa matriks P = 2 1 3 2 , Q= (a) Find Carikan (j)
the value of k, if matrix P has no inverse nilai k, jika matriks P tidak mempunyai songsang,
(ii) the inverse of matrix Q. matriks songsang bagi Q. (a) Using matrices, calculate the value of x and the value of y that satisfy the following simultaneous equations : Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : 2x + y = 4 3x ─ 2y = 13 [ (a) (i) k = – 3 (ii)
78) a)
TOPIC: MATRICES
b)
75)
3 4 1 2 a) Find the inverse matrix of G =
Hence, calculate the values of h and k which satisfy the following matrix equation.
1 1 6 8
b) Using matrices, calculate the values of x and y which satisfy both of the following equations. 3x – 4y = 7 x + 2y = 4 1 2 4 Inverse G 10 1 3 (a)
1 1 6 8 Find the inverse matrix of
inverse F (a)
1 1 8 2 6 1
h 2 k = 15 (b) h = 1/2 , k = 3/2
(b) x 3 , y = ½
79)
76) a)
h
k
4 5 3 2 Given that the inverse matrix of 5 7 4 7 , 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 (a) h = – 2/7 , k = 3/7 (b) x = 6
y=–5
77)
m 1 10 n and a) Given that F is the matrix 1 n 1 the inverse matrix of F is 10 10 5 find the value of m and n. b)
Hence, calculate the values of x and y which satisfy the following matrix equation.
x 4 y 6 F = (a) ) m = 5 , n = 4 (b) x = 1 , y = –1
is
1 1 3 1 a) Given that M is the matrix 1 1 m n 1 , and the inverse matrix of M is k Find the values of k, m and n b)
hence, calculate the values of x and y which satisfy the following matrix equation.
x 3 y 5 M = (a) k = 4 , m = –1, n = 3 (b )
x=2
y= 1
Inverse M (a)
80)
5 4 3 2 and F = Given that E =
a) b) c)
4 8 k 10 ,
4 0 0 4 Find the value of k where EF = State the inverse matrix of E Hence, using matrices, find the value of x and y which satisfy the following matrix equations.
5 4 3 2
x 6 y = 4
1 2 4 Inverse E 2 3 5 (c) x=2 ,y = – 1 (a) k = 6 (b)
81) a)
b)
2 1 3 4 is The inverse matrix of 4 1 p 2 , find the value of m and p. m Using matrices, calculate the value of x and y which satisfy the following simultaneous linear equations. 2x + y = 4 3x – 4y = 17 (a) m = – 1/11 , p = – 3
(b) x = 3
y= –2
82)
3 4 1 2 a) It is given that matrix M =
,
find the inverse matrix of M b) Using matrices, find the values of x and y which satisfy the following equations. 3x + 4y = –11 x + 2y = –7
1 2 4 2 1 3
(b) x= 3 , y = – 5
1 2 2 3 83) Given that the inverse matrix of 1 3 1 n 2 m . is (a) (b)
Find the value of m and of n. Hence, using matrices, calculate the values of d and e in the following simultaneous linear equations : 2d + e = 7 –2d – 3e = –1 (a) m = – 4 , n = 2
(b) d =
(a) Given that matrix
85)
2
and matrix
Diberi matriks
1 3 2 4
(a) m = 10 , n = 4
x 8 y 6
(b) x = 5 y = – 1
1 2 .
s
1 3
2 4
P
dan 2
Q
matriks
s
1
1 2 .
Find the value of s such that
1 0 PQ 0 1
.
Cari nilai s dengan keadaan 1 0
PQ (b) Using matrices, calculate the value of x and y that satisfy the following matrix equation :
2 4
1
Q
e=5,y=–3]
1 1 3 n 3 1 0 2 4 2 1 0 1 , 84) (a)Given m find the value of m and n.
1 3
P
0 1
.
(b) Write the following simultaneous linear equations as a matrix equation. Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks.
x 2y 4 3 x 4 y 11 Hence, using matrices, calculate the value of x and of y.
Seterusnya, dengan menggunakan kaedah matriks, hitung nilai x dan y . (a) s = 3/2
(b) x = 3 , y = ½
2 5
1 4 86. (a)The inverse matriks of is r 5 1 p 1 2 Find the value of p and of r 2 5 1 4 Matriks songsang bagi 1 r 5 ialah p 1 2 Carikan nilai p dan nilai r (b)
Using matrices, calculate the value of x and of y that satisfy the following simultaneous liner equation : Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan linear serentak berikut : −2x + 5y = 12 −x + 4y = 3 [ (a) p = – 3 r = 4 (b) x = – 11 , y
=–2]
87. (a) Given
4 6
k
1 2
2 6
1 1 0 p 0 1 , find the value of k and of p.
Diberi
4
k
6
1 2
2 6
1
1 p 0
, tentukan nilai k dan nilai p.
0
(b) Hence, by using matrices, calculate
the value of x and of y that satisfy the following simultaneous linear equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut
4 x y 13 6 x 2 y 22 [ (a) k = ½ p = 4 (b) x = – 2 y = 5 ]
1
88. (a) It given that matrix P = n 1 3 2 , find the value of n if matrix P not have inverse matrix. If n = 5,find the inverse matrix of P.
n 1 3 2 , cari Diberi matriks P = nilai n jika matriks P tidak mempunyai songsang . Jika n = 5, cari matriks songsang bagi P. (b) Hence, by using matrices, calculate the value of x and of y that satisfy the equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai v dan nilai w yang memuaskan pesamaan itu.
5 1 x 7 3 2 y 1 1 2 1 13 3 5 [ (a) n = – 3/2 inverse P =
Name : ___________________________ ____ Form : ___________________Matirces 2_12 89. M is a matrix 2 2 such that M 4 3 1 0 7 5 0 1
M ialah satu matriks 2 2 dengan
4 3 7 5 keadaan M
1 0
0 1
(a)
Find the matrix M. Carikan matriks M. (b) Write the following simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut
dalam
bentuk persamaan matriks . 4x + 3y = 23 7x − 5y = −11
x=1,y=–2]
Hence, by using matrices, calculate the value of x and of y that satisfy the equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan itu.
[ (a) Inverse M = 1 5 3 41 7 4
(b) x = 2 , y = 5 ]
3 2 8 4 and 90. Given that matrix A = 4 2 k 8 h such that AB matrix B = 1 0 0 1 . = 3 2 8 4 Diberi bahawa matriks A = dan
4 2 8 h dengan matriks B = k
keadaan
1 0 0 1 . AB = (a)
Find the value of k and value of h. Carikan nilai k dan nilai h.
(b)
Using matrices, find the value of x and of and of y that satisfy the following simultaneous linear equations. Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : − 3x + 2y = 6 − 8x + 4y = 4
[ (a) k = ¼ h = – 3
(b) x = 4 , y = 9 ]91.
2 k 2 3 and Given that matrices P = 2 1 3 2 , Q= 2 k 2 3 Diberi bahawa matriks P = dan
2 1 3 2 , Q= (a)
Find Carikan (i) the value of k, if matrix P has no inverse nilai k, jika matriks P tidak mempunyai songsang, (ii) the inverse of matrix Q. matriks songsang bagi Q.
(b) Using matrices, calculate the value of x and the value of y that satisfy the following simultaneous equations : Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : 2x + y = 4 3x ─ 2y = 13
[ (a)(i) k = ¼ (ii) h = – 3 (b) x = 3 , y = – 2 ]92.
1 4 6 6 1 3 (b) x = – 2 y = 3 ] [ (a) M =
Given M is a 2 2 matrix such that Diberi M ialah satu matrik 2 2 dengan keadaan
3 6 1 4 M
Write
the
following
simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut
find the value of x and of y.
Diberi
Carikan matriks M.
dalam
bentuk persamaan matriks . 3x + 6y = 12 x + 4y = 10
value of x and of y that satisfy the equations. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan itu.
5 y 7 6 x
6 4 7 5 =
1 0 0 1 , carikan nilai x dan nilai y. (b) Write the following simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut
dalam
bentuk persamaan matriks . 6v – 4w = 4
Hence, by using matrices, calculate the
6 4 7 5 =
1 0 0 1 ,
1 0 0 1 =
(a) Find the matrix M. (b)
93. (a)
5 y 7 6 Given that x
7v – 5w = 7 Hence or otherwise find the values of v and w . 6 4 v 4 7 5 w 7 [ (a) x = – ½ y = 4 (b) v=–4,w=–7]
94. Given M is a 2 2 matrix such that Diberi M ialah satu matrik 2 2 dengan keadaan
3 6 1 4 M
1 0 0 1 =
(a) Find the matrix M.
5 m
1 n 95. Given A = and the inverse n 8 1 1 5 matrix of A is 2 5 m
1 n Diberi A =
Carikan matriks M.
1
(b) Write the following
ialah 2
simultaneous linear equations as a matrix equation . Tuliskan persamaan serentak berikut dalam bentuk persamaan matriks . 3x + 6y = 12 x + 4y = 10
(a) (a)
8
1
5
n
Find the value of m and n. Carikan nilai bagi m dan n.
(b) Write the following simultaneous linear equations as matrix equation: (b) Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks: 5x + 8y = 4
Hence, by using matrices, calculate
the
x + 2y = –2
value
of x and of y that satisfy the equations.
[ (a) , y = 3]
dan matriks
songsang bagi A
Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan pesamaan itu. 3 6 x 12 1 4 6 6 1 3 (b) 1 4 y 10 x = – 2
Hence, using matrices, calculate the value of x and of y. Seterusnya, dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y 5 8 x 4 1 2 y 2 [ (a) m = 8 n = 2 (b) x = 12 , y = – 7 ]
2 k 96. Given that matrices P = 2 3 and 2 1 3 2 , Q= 2 k 2 3 dan Diberi bahawa matriks P = 2 1 3 2 , Q= (a) Find Carikan (j)
the value of k, if matrix P has no inverse nilai k, jika matriks P tidak mempunyai songsang,
(ii) the inverse of matrix Q. matriks songsang bagi Q. (a) Using matrices, calculate the value of x and the value of y that satisfy the following simultaneous equations : Dengan menggunakan kaedah matriks, hitungkan nilai x dan nilai y yang memuaskan persamaan linear serentak berikut : 2x + y = 4 3x ─ 2y = 13 [ (a) (i) k = – 3 (ii)
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