School of Engineering – UCSI UNIVERSITY EE107 (Jan-April 2013)

1

Mock Test1 (XY Feb 2013): Questions & Solutions

Q1 Consider the matrix

dc

baA

.

(a) Find the value of , as a function of a, b, c, d and , that makes A

nonsingular.

(0.5 marks)

(b) Find the value of that makes A nonsingular.

(0.5 marks)

(c) If = , find the value of a that makes A singular.

(0.5 marks)

Q1. (a).

bc

ad

(0.5 mark)

(b).

ad

bc

(0.5 mark)

(c). abc

d

(0.5 mark)

Q2. Solve the linear system by (a) Gauss elimination and (b) Cramer Rule:

BAX

3

2

1

211

012

131

z

y

x

Solution: Using Gaussian Elimination Method:

3

2

1

211

012

131

MatrixAugmented

233 2 RRR 122 2RRR 233 5 RRR

4

2

1

410

012

131

4

0

1

410

250

131

20

0

1

1800

250

131

School of Engineering – UCSI UNIVERSITY EE107 (Jan-April 2013)

2

9/102018 zz

9/40)9/10(25

025

yy

zy

9/111)9/10()9/4(3

13

xx

zyx

(2 marks)

(b) Using Cramer’s Rule:

A

Ax 1

213

012

131

1A

1110112

3120

13

1211 A

9:)( AthatknowsaFrom

9/11x

A

Ay 2

231

022

111

2A

400421

1120

31

2212 A

9/4y

A

Az 3

311

212

131

3A

10112111

121

31

223

31

2113 A

9/10z

(2marks)

Q3. Given

22

25A ; find the eigenvalues and corresponding eigenvectors.

The eigenvalues are determined via characteristic equation:

0 IA

School of Engineering – UCSI UNIVERSITY EE107 (Jan-April 2013)

3

016

067

042510

0)2)(2(25

022

25

00

0

22

25

010

01

22

25

2

2

61 12

The corresponding eigenvectors are determined via:

0 XIA

when 61 :

0)6( XIA

042

21

010

01)6(

22

25

y

x

y

x

02 yx

kykxLets2

1

2/1

1:, isreigenvectotheHence

when 12 :

0)1( XIA

012

24

010

01)1(

22

25

y

x

y

x

024 yx

kykxLets 2

2

1:, isreigenvectotheHence

(3 marks)