King Fahd University of Petroleum and Minerals
Department of Mathematical Sciences
Math260 (A. Farhat)
Midterm Exam
Summer, 2001 (003)
July, 23 2001
Instructions:
Answer all questions. Show all your work.
Cheating will be met with harsh punishment
NAME:
______________________________
I.D. No.: _______________
Sec. No.: _______
No. 
Points 
Score 
1 
18 

2 
5 

3 
5 

4 
6 

5 
3 

6 
5 

7 
5 

8 
4 

9 
5 

10 
18 

11 
4 

12 
12 

13 
10 

Total 
100 
1. Given the
differential system
a) Find the eigenvalues.
b) Find the associated eigenvectors.
c) Find all solutions of the system.
d) Give the general solution of the system.
e) Give the general solution in scalar form.
f) Find the solution when
g) Can you solve the system directly? If yes, find the solution.
2. Let
where A is a
is also a solution.
3. Show whether the
mapping
is linear or nonlinear.
4. Let A be an
a) Describe two methods to
solve the linear system
b) Let the rank of A be
If you think that the system is consistent, how many solutions will it have?
Zero, one, or infinitely many?
c) If the system is homogeneous and the rank
of A is
5. If the square of the
length of a vector x is 7, find
6. If the angle between
the vector
7. Let A be a
where
a) Find the determinant of A.
b) Under what condition is A invertable?
8. Find a
9. Let
10. Let
a) Write the system in matrix form.
b) Obtain two realvalued solutions from the real and imaginary parts of the complex solution.
c) Give the scalar form of the imaginary part of the complex solution.
d) Verify that the scalar form in part (c) satisfies the second differential equation.
e) Give the general solution of the system.
11. An
12. Let
a) Find another eigenvector
b) Give two solutions of the system.
c) Give the general solution of the system.
13. Answer the following questions with (T)rue or (F)alse.
________ 1) If
x and y are solutions to the nonhomogeneous system
________ 2) The principle of superposition states that if x and y are two solutions to a homogeneous system, then any linear combination of x and y is also a solution to the system.
________ 3) Matrix multiplication is associative.
________ 4) Matrix multiplication is always defined for square matrices.
________ 5) A
matrix is symmetric if
________ 6)
________ 7) The trace of a square matrix is the sum of its diagonal elements.
________ 8)
________ 9) An
________ 10) A linear system with less equations than unknowns is always consistent.