Assignment 8   427K-H  due Wednesday, March 31

Remember there is an exam on March 31  in your TA section
.
1. Find the general solution of the equation z' = Az  where A is the

matrix  [-3, 3;1, -5].


2.  The eigenvalues of the matrix [-9 -3 -7;3 1 3;11 3 9]  are 1, 2 and -2.

Find the eigenvectors.


3. Suppose a 2x2 matrix A has eigenvalue 2 + i with eigenvector [ 3; 2 - i].

Find a real basis for the solution space of  z' = Az where z is a vector

z = [x ; y]. Now find the solutions with x(0) = 1 and y(0) = 1
 
  

4.Consider the 2x2 system of equations z' = Az  where the matrix A

depends on an unknown parameter b .  A = [-1,b^2; -1, 1]. Find the eigenvalues 

and eigenvectors and give a real basis for the solution spaces as they

depend on b. Pay some attention to b = 0. 


5.We have not learned how to solve  system z' = Az where A = [1 4; -1  -3].

Find the general solution using the assumption that there are two solutions,

one of the form e^rt[v_1;v_2]  and another of the form 

e^rt([ w_1;w_2]+ t [v_1;v_2]). Here the vectors v and w do not depend on t.


Extra credit (due Thurday April 1).  Prove that if A is a 2x2 matrix, then

det A = r s   and trace A = r + s, where r and s are the two eigenvalues of A.