Department of Mathematics
Center for Foundation Studies, IIUM
Semester I, 2011/2012
SHE 1114 (MATH II)
TUTORIAL 5
CHAPTER 5: SYSTEM OF EQUATIONS
Page
Questions
Number
9.1
255 -256
13, 17, 39
9.2
262 -263
3, 15, 61
9.3
270 - 271
15, 67
9.4
278
7, 19
9.5
281
9
9.6
286
9, 19
Chapter 9 Review
288
9, 15
*Required Textbook: Parveen Kausar Yacob et al , Mathematics for Matriculation: Algebra,
2nd Edition, (2011) Cengage Learning Asia Pte Ltd.
Section
EXTRA QUESTIONS:
1
 5  1
 8 4 
 , B = 
 and C =
If A = 
 2 3 
 4  6
a) show that (A+B)+ C = A +(B+C)
b) find 2A – 3B +2C
2
3
 1 2 3


Let A =  0 4 1  and B =
 1 1 3


AB = BA and find A-1.
  2 3


 5 3
Ans:
𝟑𝟎
(
−𝟔
−𝟖
)
𝟑𝟎
 11  3  10 


0
 1  show that
 1
 4 1
4 

Ans: B
 7 16  10 
1  2  6 




Given A =  1  1 2  and B =  5 11  8  , find AB and hence
 1  1
2  3  1 
1 



write down the inverse matrix A-1. Use this matrix to find the values of x. y
 x  1
   
and z given that A  y  =  2 
 z  3
   
Ans: x =3, y =1, z =0
1
4
Find the inverse of the matrix by using row operation method
2  1 3 


 5 4  3  . Hence solve the system of linear equations
3  2  1


2x – y + 3z = -25
5x + 4y – 3z = -1
3x – 2y – z = -17
Ans: x = - 5, y = 3, z = - 4
5
1
0 
 1


2  and A = -5, find the value of x.
If A =   3 x  2
 0
 2 x  1

6
Solve for x
1
1|=-5
5
−1
−2
Find the value of the determinant |𝑥 + 2𝑎 𝑦 + 2𝑏
𝑎
𝑏
known that
𝑥 𝑥
a) |
|=-48
−4 4
7
𝑥
|1
𝑎
8
𝑦
2
𝑏
𝑥 1
b) | 2 3
−1 3
𝑧
3| =4
𝑐
Ans: x=2
Ans: a) -6 b) -1/4
−3
𝑧 + 2𝑐| if it is
𝑐
Ans: 4
Solve the system of equation using Cramer’s Rule
𝑥+𝑦 =8
𝑥−𝑦 =2
a)
{
c)
𝑥 + 𝑦 − 𝑧 = −6
{ 3𝑥 − 𝑦 + 4𝑧 = 5
𝑥 + 5𝑦 − 3𝑧 = −24
3𝑥 − 𝑦 = 4
5𝑥 + 4𝑦 = 35
b)
{
d)
𝑥 + 5𝑦 − 𝑧 = 5
{ 2𝑥 − 5𝑦 + 𝑧 = 1
−3𝑥 + 5𝑦 − 5𝑧 = 1
Ans: a) x = 5, y = 3 b) x = 3, y = 5; c) x = -1, y = -4, z = 1; d) x = 2, y = 2/5, z = -1
9
Solve the system of linear equations
2𝑦 + 𝑧 = 4
𝑥+𝑦 =4
3𝑥 + 3𝑦 − 𝑧 = 10
by using elementary row operations.
2
Ans: x = 3, y = 1, z = 2
10
11
2  1 3 


Find the cofactor and adjoint of the matrix  5 4  3  . Hence, find
3  2  1


the inverse matrix and solve the system of linear equations
2x – y + 3z = -25
5x + 4y – 3z = -1
3x – 2y – z = -17
Ans: x = -5, y = 3, z = -4
Given the system of linear equations
3𝑥 + 2𝑦 = 45
𝑥 + 𝑦 + 𝑧 = 40
4𝑥 − 𝑧 = 0
a) Write down the above equations as a matrix equation in the form
AX = B
b) Obtain the adjoint matrix of A and the determinant |𝑨|.
Hence, find A-1. By using A-1, solve for x, y and z.
Ans: x = 5, y = 15, z = 20
12
Given the system of linear equations
𝑥−𝑦−𝑧=1
2𝑥 − 𝑦 + 𝑧 = 2
2𝑥 − 2𝑦 − 𝑧 = 3
a) Write down the above equations as a coefficient matrix A.
b) Find the adjoint matrix of A and A-1 and hence solve for x, y and z.
Ans: x = -1, y = -3, z = 1
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