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TUTORIAL MATRICES AND SYSTEMS OF LINEAR EQUATIONS
1.
 5
1
1  1
1  3 0 

Given that A = 
, B =
,C= 4
 3 and D =



0 2 
3  6  2
 5 2 
3 4 0
 2 4 3


1 3 2
evaluate the following;
2.
3.
(a)
BC + 2A
(b)
D2 – CB
(c)
A – 3(BC + A)
5 0 0 
Given two matrices A = 1 8 0  , B =
1 3 5 
 2 0 0 
  1 5 0 


 1 3 2 
(a)
Determine if AB = BA
(b)
Find m and n such that A = mB + nI3.
 2 3
The matrices A and B are given as A = 
, B=
 1 1
a b 
 1 1 where a and b


are real numbers. Find the diagonal matrix D such that ADA-1 = B.
4.
1 1 1 


Given that matrix P = 1 0 1  , find the matrix A such that
 0 1 2 
 2 0 0 


P AP =  0 2 0  .
 0
0 4 
-1
5.
Find the inverse matrix of the following using;
(i)
The adjoint method
(ii)
The Gauss-Jordan Elimination method
(a)
6.
0
2 1
 4  1  5


1  1 2 
(b)
1 1 1
Given that matrix A = 1 0 2 and B =


2 3 7
 2 4 2 
 4 5
3 

 1
1  1
4  2
6
 3  5 1  . Prove that AB = kI,


 3 1
1 
where I is an identity matrix and k is a constant.
7.
 2 1
2
Show that A = 
 is a zero of polynomial f(A) = A – 5A + I.

5
3


Hence find the matrix A-1 and A3
8.
1 2 1 
Given that matrix A = 1  1 0 , find the matrix A2. If A3 – A2 – 6A = I,
3 1 1 
where I is an identity matrix , find the inverse matrix of A.
9.
2  1 1 
If P = 1 0 1  and Q =
3  1 4
3  1
1
 1 5  1 , find PQ and


 1  1 1 
deduce the matrix P-1. Express the following system in its matrix form.
2x  y  z  3
x  z 1
3x  y  4 z  0
Hence solve the linear system.
10.
Solve the following systems of equation system by using,
(a)
The Inverse matrix :
x  2y  z  5
2 x  3 y  2 z  1
x  3y  2z  2
(b)
Cramer’s rule :
x  3y  2z  1
5 x  2 y  7 z  2
x  2y  z  2
(c)
The Gauss-Jordan Elimination :
2 x  3 y  4 z  1
2 x  2 y  3z  2
x  2 y  2 z  3
11.
 1 4 2 
 3 2 2


2  , find MN. Hence find M-1.
If M =  4 1 2 and N =  8 5
 2 1 5 
 2 1 3
In a supermarket, there were three promotion packages A, B and C which offer
shirts, long trousers and neckties. The number of each item and the promotion
price for each package is given in the following table:
Package
No. of
No. of
No. of
Promotional price
Shirts
trousers
neckties
(RM)
A
3
2
2
256
B
4
1
2
218
C
2
1
3
173
By using x, y and z respectively to represent the price of a shirt, a pair of trousers
and a necktie, write down a matrix equation to represent the information above.
Solve the equation to determine the price of each item.
12.
Three weight lifters, Herman, Thor and Akiro entered a weight-lifting contest in
which they could make up their total weight by adding three different sized
weights. Herman used 4 of weight X, 3 of weight Y and 5 of weight Z for a total
lift of 295kg.
Thor used 2 of weight X, 5 of weight Y and 3 of weight Z for a total lift of 295kg.
Akiro used 3 of weight X, 4 of weight Y and 5 of weight Z for a total lift of 310kg
(a)
Use the above information to determine the three separate weights by
using elementary row operation method.
(b)
If Thor replaced one weight Z by a weight Y, would this change the result
of the competition?
13.
Use the quadratic function y = ax2 + bx + c to model the following data :
x (Age of a driver)
y (Average number of automobile
accidents per day in the United States)
20
400
40
150
60
400
Use Cramer’s Rule to determine values for a, b, and c.
14.
A mixed nut company uses Cashews nuts, Macadamias nuts and Brazil nuts to
make 3 gourmet mixes. The table below indicates the weight in hundreds of
gram of each kind of nut required to make a kilogram of mix.
Cashews nuts
Macadamias nuts
Brazil Nuts
Mix A
5
3
2
Mix B
2
4
4
Mix C
6
1
3
If 1 kg of mix A costs $12.50 to produce, 1 kg of mix, B costs $12.40 and 1 kg of
mix C costs $11.70, determine the cost per kilogram of each different kind of nut.
Hence, find the cost per kg to produce a mix containing 400 gram of Cashews
nuts, 200 gram of Macadamias nuts and 400 grams of Brazil nuts.
ANSWERS
1.
(a)
 9 2
 11 3


2.
(a)
AB = BA
3.
 3 0 
 0 2


4.
1 3 3 


A =  3 5 3 
 6 6 4 
5.
(a)
7
1 
13
27 
 3
4
3
6.
7.
3 1 
A-1 = 

5 2 
8.
6 1 2
A   0 3 1  ,
7 6 4 
2
(b)
(b) m = -1
5 
10 

6 
2
1 0 0 
6 0 1 0 , k  6
0 0 1 
 31 1 2 
 22 27 12 


10 19 17 
 43 24 
A3 = 

 120 67 
 1  1 1 
A   1  2 1 
 4
5  3
1
 31 10 
33 1 


(c)
n=3
(b)
8
1 
1
30 

9
2
4
6
22 
14 

6 

9.
10.
1 0 0
PQ  20 1 0 ,
0 0 1
P
1
 1 3  1
1
  1 5  1 ,
2
 1  1 1 
(a)
z  2
(c)
1
3
x2
x3
y 1
y  5
23
21
52
z
21
z  2
z  2
y
1 0 0 
MN = 9 0 1 0 
0 0 1 
M
1
 1 4 2 
1
  8 5 2 

9
 2 1 5
X = RM 30
12.
y 1
(b)
x
11.
x3
Y = RM 68
(a)
X = 25kg, Y = 40kg, Z = 15kg
(b)
Yes, Thor lifts 320kg
5
, b  50 , c  1150
8
13.
a
14.
Cashews $12 per kg
Macadamias $15 per kg
Brazil nuts $10 per kg
$11.80
Z = RM 15