Chapter 3 Matrices

KOLEJ TINGKATAN ENAM SHAH ALAM

CHAPTER 3 : MATRICES

TOPICAL TEST

TIME : 1 hour 20 minutes

Name : …………………………………………….. Class : ………………………………..

Instruction : Answer all questions.

5 0 0  − 2 0 0 
1. The matrices A and B are given by A = 1 8 0 and B =  −1 − 5 0 
1 3 5  −1 − 3 − 2

. [ 3 marks ]
[ 4 marks ]
a) Determine whether the matrices A and B are commutative.

b) Find the real numbers m and n for which A = mB + nI, where I is the 3 x 3
identity matrix.

 1 4 − 2
2. A matrix A is given by A =  −1 1 −1 . By using elementary row operation, find

 3 0 1 

the inverse of A .
[6 marks]

3. ( a) Determine the values of m, n and p where p  0 such that A is a

symmetrical matrix.

− − − 3 [4 marks]
= ( − − ) [2 marks]
−
+ −

(b) Show that A has an inverse.

3 2
4. Given that matrix = (0 + 1 6) and | | = 60.

0 0

(a) Find the values of x.

[3 marks]

(b) By using the positive value of obtained in (a), find the inverse of
matrix A by using elementary row operations.
[5 marks]

5. The matrices and are given by

111 4 4 4

= ( ) , = (1 )

1

(a) Show that det = ( − )( − )( − ). [ 4 marks ]
(b) Deduce det . [ 4 marks ]

6. The variables , and satisfy the system linear equations

2 + + 2 = 1

4 + 2 + =

where k is a real constant. 8 + 4 + 7 = 2

a) Write a matrix equation for the system of linear equations. [1 marks]

b) Reduce the augmented matrix to row-echelon form, and show that the system of
linear equations does not have a unique solution. [6 marks]
c) Determine all the values of for which the system of linear equations has

infinitely many solutions, and find the solutions in the case when is positive.
[6 marks]
d) Find the set of values of for which the system of linear equations is

inconsistent. [2 marks]