Name:
CSU ID:
Version 1, Exam 1, Math 369
September 25, 2013
Show all work to receive full credit.
Part A
1. Given the matrix A and vector ~b below, find the following: (a) RREF of
[A|~b] (the intermediate elimination steps need not be defined); (b) the
number of free variables and what they are; (c) homogeneous solution
of A~xH = ~0 with ~xH written in vector form; and (d) the general
solution of A~x = ~b written in vector form with the particular solution
and the homogeneous solution(s) in separate vectors.




A=
2 −3
0
1
0
−4
6
3
0
1
8 −12 −9 −2 −2
2 −3 18 13
1




,


~b = 


14
−4
−12
138





2. (a) Given A below, if the (1,1) position is to remain unchanged, define
the operator, E1 , that leaves row 1 unchanged and eliminates
column 1 from the second row to the third.


2 −3
4


9 −13 
A =  −4
8 −21
29
(b) Given that the LU decomposition of A is



1
0 0
2 −3
4



1 0  0
3 −5 
A = LU =  −2
4 −3 1
0
0 −2
and


37

~b = 
−113


253
solve A~x = ~b using forward and backward substitution. The
answer should be defined with fractions, if necessary.
3. Find a vector w
~ consisting of integers, such that it is perpindicular to
the following vectors




~v1 = 
−1
2
−1
0



,





~v2 = 
−3
9
−3
2



,





~v3 = 
−1
−10
2
−8





Name:
SI:
Version 1, Exam 1, Math 369
September 25, 2013
Show all work to receive full credit.
Part B
4. Refer to the matrix A in problem 2.
(a) Find the cofactor C2,3 .
(b) Using the LU decomposition, find det(A).
(c) Using the information from (b), if B is the matrix produced by
flipping columns 1 and 3 of A, what is det(B)?
5. Consider the vectors ~v1 and ~v2 in ]3.
(a) Find the angle between ~v1 and ~v2 . Is it acute or obtuse?
(b) Find proj ~v1 ~v2
(c) Find the length of proj~v1 ~v2 .
(d) Find w
~ such that (1) ~v1 and w
~ are orthogonal and (2)~v2 =
proj~v1 ~v2 + w
~