King Fahd University of Petroleum and Minerals

Department of Mathematical Sciences

Math-260 (A. Farhat)

Midterm Exam

Summer, 2001 (003)

July, 23 2001

Instructions:

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  and  be two solutions of the system

where A is a  matrix. Show that the linear combination

is also a solution.

3.   Show whether the mapping

is linear or nonlinear.

4.   Let A be an  matrix.

a)  Describe two methods to solve the linear system

b)  Let the rank of A be . Will the system be consistent? Why?
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 , how many arbitrary variables will be involved in the solution of the system?

5.   If the square of the length of a vector x is 7, find .

6.   If the angle between the vector  and a vector v with length 4 is , find . Simplify your answer.

7.   Let A be a  matrix satisfying:

,       ,       ,
where  are real number.

a)  Find the determinant of A.

b)  Under what condition is A invertable?

8.   Find a  matrix that reflects vectors in the xy-plane across the -axis.

9.   Let ,   ,  and  . Express z as a linear combination of x and y. That is, find  and  such that  .

10. Let  be a complex valued solution to the system of differential equations

a)  Write the system in matrix form.

b)  Obtain two real-valued 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  matrix is said to be singular if its determinant is zero. Give two consequences when a square matrix is singular.

12. Let  be an eigenvalue of multiplicity 2 for a linear homogenous system of differential equation with coefficient t matrix . Let  be an eigenvector associated with .

a)  Find another eigenvector  associated with .

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 , then their difference is a solution to the homogenous equation

________  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  matrix is invertible if its row reduced echelon form is equivalent to .

________  10)   A linear system with less equations than unknowns is always consistent.