Download 1 - University of Malta

UNIVERSITY OF MALTA
FACULTY OF INFORMATION & COMMUNICATION TECHNOLOGY
DEPARTMENT OF ARTICIAL INTELLIGENCE & DEPARTMENT OF MATHEMATICS
B.Sc. (Hons.) ICT Evening Degree / Diploma in ICT
May / June 2008 Assessment Session
4th June 2008
MAT1092: Mathematical Methods for ICT
18:00 – 19:30
This paper contains two sections. You are to attempt two questions from each section.
Section A – Matrices
1)
a.
If A is the matrix formed from the left-hand side of the equations, while B is the matrix formed
from their answers, determine:
i. A.B
2x + 9y – 5z = -7
z – 2y = 2
2z – x – 3y = 3
ii. At
iii. A-1
b.
[2 marks]
[2 marks]
[2 marks]
iv. Bt
[2 marks]
v. Bt.A-1
[2 marks]
vi. |A|
[5 marks]
vii. The values of x, y, and z using A-1
[5 marks]
viii. Confirm the values in (vii). by employing Cramer’s Rule
[5 marks]
Consider the matrix
5
-3
3
2
6
-3
4
2
1
and determine:
i. the Eigenvalues
[9 marks]
ii. the corresponding Eigenvectors
[12 marks]
iii. whether it is diagonizable or not (show working)
[4 marks]
Total marks for question one: 50 marks
Page 1 of 3
2)
If M =
1 2 1
6 -1 0
-1 -2 -1
, determine:
i. M.Mt
[3 marks]
ii. |M|
[7 marks]
iii. the matrix N, where N = Mt.M – 2I, where I is the identity matrix
[10 marks]
iv. the rank of M-1, Mt, M.Mt, Mt.M, and N
[10 marks]
v. the eigenvalues and eigenvectors of M
[10 marks]
vi. Hence, or otherwise, determine M5
[10 marks]
Total marks for question two: 50 marks
3)
a.
Q=
½
2
- /2
½
½
2
/2
½
2
/2
0
-2/2
i. Work out Q.Qt.
[5 marks]
ii. Hence determine Q-1.
[5 marks]
iii. Find |Q|.
[5 marks]
b.
If A =
1 2 -1
2 1 1
5 4 1
and B =
1 -1 0
-1 1 1
-1 1 2
i. Calculate A.B
[7 marks]
ii. Determine the eigenvalues of A
iii. Show that two of the eigenvectors of A are
iv. Find the third eigenvector
1
-1
-1
and
v. Is A diagonalizable? And if so, work out the value of A7
-1
1
1
[7 marks]
[7 marks]
[7 marks]
[7 marks]
Total marks for question three: 50 marks
Page 2 of 3
Section B – O.D.E.
4) Solve the following differential equations:
a.
(x + y – 2) dx + (x – y + 4) dy = 0
[15 marks]
b.
dy
[15 marks]
c.
y” + 3y’ – 4y = 18e2x
[20 marks]
/dx + 2y/x = -2x4/3
Total marks for question six: 50 marks
5) Prove that the following FODE are not exact, determine their Integrating factor, and eventually prove
that they can be made exact:
a.
2xy.y’ + x2 = -3y2
[15 marks]
b.
x.y’ + y +x2y = 0
[15 marks]
c.
y’
[20 marks]
/x + 2y = -1
Total marks for question four: 50 marks
6) Find the general solutions to the following SODEs:
a.
d2y/dx2 –
/dx – 12y = 14sin2x + 18cos2x
b.
d2y/dx2 – 3 dy/dx + 2y = 3e-x
[15 marks]
c.
y” – 2y’ + y = 2x2 + 2x -7
[20 marks]
dy
[15 marks]
Total marks for question five: 50 marks
Page 3 of 3