Chapter 4 Matrices
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CHAPTER 4 MATRICES
1.
Given that P = P =
), , Q = Q = ( and PQ = PQ = ( ( ) )
k and n , (a) find the value of k and (b) hence, using matrices, solve solve the following following simultaneous equations: 2x – y = 7 – 3y = 5x – y = 16 – 6y =
2.
(a) Express
) ( )as a single matrix. (
(b) Using matrices, matrices, solve the following following simultaneous equations: 5x – y = 7 – y = 4x + x + 3y 3y = = – 2
3.
(a) If K = K =
), find the matrix L. and KL = ( ( )
(b) Hence, using matrices, solve solve the following following simultaneous equations: 2x – y = 1 – 5y = 4x – y = – 1 – 7y =
4.
(a) Find the inverse matrix of
( ).
(b) Hence, using matrices, solve solve the following following simultaneous equations: y = 9 2x – – 3y = 5x – y = 19 – 4y =
5.
Given that S = S =
( ),
m such S has no inverse, such that S has (a) find the value of m (b) if m = – 5, hence, using matrices, solve the following simultaneous equations: 4x – y = 7 – 3y = 7x + x + my = my = 12
6.
Given that P = (
),
(a) Find the inverse matrix of P , (b) Hence, using matrices, solve the following simultaneous equations: 6x – 3y = 0 4x – 5y = – 12 7.
) () () , it is found that
In solving the matrix equation(
() ( )() .
(a) Find the values of m , p , q , r and s . (b) Hence, find the value of x and y .
8.
(a) If given T = (
), find the inverse of matrix T .
(b) Hence, calculate the value of x and y that satisfies the following equation: 7x + 2y = 3 6x + 3y = 0
9.
Given that the inverse matrix of
( ) is K ,
(a) find the matrix K, (b) hence, using matrices, solve the following simultaneous equations: 4x – y = 11 6x – 5y = 27
) and the inverse matrix of K is ( ).
10. Given that K = (
(a) Find the value of m and n . (b) Hence, using matrices, calculate the value of x and y that satisfies the following equation:
K () ()
ANSWER: 6 12 15 5
2 3 5 4
3
1
2
4) a) Let A as the inverse matrix of
4 8 15 5
1) a) Q =
3
1
1 6
3
3 5
2
A =
4 7 5
= k = 3, n = – 6
1
2 3 x 7 5 6 y 16 3 7
2 16
1 3
1 5
5 4
2) a)
b)
x y 19 0
1 19 0 1
3 9
0
1
5 1 x 7 4 3 y 2
5) a) If S has no inverse, 4m + 21 = 0 m = 5.25
4 3 x 7 7 5 y 12 b)
x 5 1 y 20 21 7
b) 1 7
5 2
x = 1, y = – 2
7 14 20 4 1
5 30 12 4
5
2
L = 1 7 5
2 5 x 1 4 7 y 1 b)
7 6 4
6
5 18 4 1
3
6
=
2
6 3 x 0 4 5 y 12
=
1
3
A =
x y
4 12
6) a) Let A as the inverse matrix of P.
3) a) L is the inverse matrix of K .
3 7
x = 1, y = – 1
1
6 4
2 19
19
=
4 7 5 1
x = 3, y = – 1
0
=
x 1 3 y 19 4
2
2 3 x 9 5 4 y 19
x = 2, y = – 1
3 4
3
=
b)
x 1 6 y 3 5
2
5 1
2 1
x = – 2, y = – 1
b)
x 1 5 18 4 y x = 2, y = 4
3 0
6 12
.
x 5 7 1 1 y 4 2 10 28 1 7) a)
10) a) Let A as the inverse matrix of K . m = – 4, – 4n = 6 – 34 n=7
5 7 1 18 4 2 1 1
= m =18, p =5, q = – 7, r = 4, s = – 2
x 1 5 7 1 y 18 4 2 1 1
9
9
x = –
, y =
x = – 1, y = 2
3 2 7 21 12 6 1
A =
3 2 7 9 6 1
= 2
x 3 3 y 0
b)
x y
3 2 3 7 0 9 6 1
x = 1, y = – 2
5 20 6 6 1
1
4
9) a) K =
5 14 6 1
1
4
=
4 1 x 11 6 5 y 27 b)
x 1 5 14 6 y x = 2, y = – 3
4 3 x 2 2 7 y 16 3 2 x 1 7 34 2 4 16 y
8) a) Let A as the inverse matrix of T .
7 6
A =
b)
b) 1
3 7 34 2 4 1
1 11
4 27
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