Exam #1, Math 205B (Linear Algebra)
This take-home exam is due at class time on Monday, October 8. (Sooner is fine.) You may consult the
textbook (or any other book) and any class notes and handouts, but please do not discuss any details of
this exam with anyone except me! Please sign the bottom of this sheet and turn it in with your exam.
You may ask me questions about the exam, but I reserve the right to give unsatisfying answers. Please show
all work (though you are encouraged to check your answers on MATLAB or a calculator).
1. (8 points) Find the angle between the vectors ļ£«
ļ£¬
ļ£
1
1
1
1
ļ£¶
ļ£·
ļ£øand ļ£«
ļ£¬
ļ£
1
1
0
1
ļ£¶
ļ£·
ļ£ø.
2. (18 points) Solve the system ļ£«
ļ£
112
113
191
ļ£¶
ļ£øļ£«
ļ£
x1
x2
x3
ļ£¶
ļ£ø=ļ£«
ļ£
3
7
15
ļ£¶
ļ£øby any method (by hand).
3. (20 points) Find Aā1(by hand) if A=ļ£«
ļ£
1 2 3
2 3 1
3 1 2
ļ£¶
ļ£ø.
4. (30 points) (a) Find the LU factorization of A=ļ£«
ļ£
1 4 3
2 9 8
3 16 18
ļ£¶
ļ£ø.
(b) Use your answer to (a) to solve A~x =ļ£«
ļ£
7
13
15
ļ£¶
ļ£ø.
(c) Find Lā1and Uā1. Please show work for at least one of these.
(d) Use your answers to (c) to compute Aā1.
5. (12 points) Let Aand Bbe square matrices of the size, say nĆnfor some nā„2, and let Odenote the
nĆnzero matrix (all of the entries of Oare zero). Assume throughout this problem that AB =O.
(i) Show by an example that BA need not equal O. (You should be able to find a 2 Ć2 example, though
you are welcome to give a larger example instead.)
(ii) If Ais invertible, must BA =O? Explain. What if Bis invertible?
(iii) If Aand Bare symmetric (i.e.,A=ATand B=BT), prove that BA =O.
6. (12 points) Suppose that Ais a 2Ć2 matrix such that Aā1=AT. Prove that Amust be either a rotation
matrix or a reflection matrix.
I affirm that I did not receive help from another person in doing this exam, nor did I give help
to another student in the class.
(signed)