Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
Quiz 7 – Friday, October 1, 1999
NAMES
1) Do the following by hand (clearly show your work). Let A =
(a) Find the eigenvalues of A:
Ã
1 2
¡1 4
!
:
(b) Choose one of the eigenvalues in (a) and …nd its eigenspace.
2)" Suppose
the"output
Maple’s eigenvects(A); command
(" that
#)#
(" from
#)#
¡3
1
is 0; 1;
; 7; 1;
: Find A: If you use Maple, be sure to tell me what you
1
2
did.
3) True or False. Please put T or F to the left of the question. Let A be an n £ n matrix.
a) A has n eigenvalues.
b) A has n linearly independent eigenvectors.
c) If dim(N (A)) = 2; then 0 is an eigenvalue of A:
d) If the elements of each row add to 1, then the vector consisting of all 1’s is an eigenvector
of A:
e) A can always be written as A = SDS ¡1 where D is a diagonal matrix.
f) dim(N (A)) + dim(R(A)) = 2n:
g) If det(A) 6= 0; then A has an inverse.
h) If dim(N (A ¡ 2I)) = 0; then 2 is an eigenvalue.
4) First …nd the eigensystem of A =
izable.
Ã
2 3
0 2
!
; and then determine whether A is diagonal-