

EE441
Dr. Golomb
2:00 p.
m.
FIRST
MIDTERM
September
28,
2006
Put your name, and all answers to be graded, on the Answer Sheet.
BE ACCURATE!
(
332
)
A. Let
M
=
1
1
 2
, a 3 x 3 real matrix.
310
Find each of the following:
1. The determinant
of
M,
IMI.
2. The trace of
M, Tr(M).
3. The characteristic
polynomial of
M,Pm(>"')'
4. The characteristic
roots (eigenvalues) of
M,
{AI,
A2,A3}'
5. Linearly independent eigenvectors {aI,
a2, a3}
corresponding to these eigenvalues.
6. A nonsingular matrix
P
such that
pI M P
= A,a diagonal matrix.
7. The diagonal matrix A of problem 6.
8. The inverse,
pI,
of the matrix
P
in problem 6.
(
112
)
B. Let
A
=
2
2
4
, a 3 x 3 real matrix.
336
9. What is the
domain
of
A?
10. What is the
nullspace
of
A?
11. What is the
rangespace
of
A?
12. What is the
order
of
A?
13. What is the
rank
of
A?
14. What is the
nullity
of
A?
(For problems 9., 10., and 11., you can describe the spaces involved by exhibiting
a
basis for each one.)
C.
Let
F7
= {O,1,2,3, 1, 2, 3}
be the field of seven elements, and let
V
=
F.f.
Find
the number of kdimensional subspaces of
V,
for
15. k
= 1
16. k
= 2
17. k
= 3
18. k
= 4
(Give your answers as expressions, and as specific numbers.)
(over)