The Open University of Sri Lanka
B.Sc. /B.Ed. Degree Programme
Pure Mathematics – Level 04
PUU2142/PUE4142 – Linear Algebra
No Book Test (NBT) – 2015/2016
DURATION: ONE HOUR
Date: 07.05.2016.
Time: 14:30h – 15:30h
ANSWER ALL QUESTIONS.
1.(a) Find the rank of the following matrix:
4 
 1 2 0


0 
3 1 1
 1 5 1 8 


 3 8 2 12 
3 1
4
3
2
(b) If A  
 use Cayley-Hamilton Theorem to express A  3 A  2 A  A  I as a
1 2
linear polynomial in A. (i.e. as  A   I where  ,    .)
(c) Determine the characteristic roots and corresponding characteristic vectors of the matrix
 2 1 1 


A where A   1 2 1 .
 1 1 2 


2.(a) Find a matrix P which diagonalizes the matrix
 4 1
A
.Verify that P 1 AP  D where D is the diagonal matrix. Hence find A 6 .

2 3
1
(b) Solve the following system using LU decomposition:
x 1  2 x2  3x3  5
2 x1  4 x2  6 x3  18
3x1  9 x2  3x3  6
 i 0 0
(c) Show that A  0 0 i  is skew – hermitian and also unitary.
0 i 0
2