EXAM
Exam 2
Math 2360–102, Summer I, 2015
June 17, 2015
• Write all of your answers on separate sheets of paper.
Do not write on the exam handout. You can keep
the exam questions when you leave. You may leave
when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 7 problems. There are 310 points
total.
Good luck!
70 pts.
Problem 1. Let A be the matrix


1
0
2

A=
 1
1

1 
.
1
2
−1
A. Find the cofactors C13 , C23 and C32 of A.
B. Find the determinant of A by expanding along a row or column. Show your
work!
C. Find the adjoint matrix adj(A) of A by calculating cofactors of A. Show
your work!
D. Using the information above, find A−1 .
40 pts.
40 pts.
Problem 2. Find the following determinant by the method of elimination, i.e.,
by using row operations and keeping track of the effect of the row
operations on the determinant. Show each row operation and the intermediate
determinants. You may do the row operations on a calculator, if you wish.
Sorry, no credit for finding it by another method.
0 1 2 6 15 15 2 4 2 Problem 3. Use Cramer’s rule to solve the following system. Evaluate the
determinats by hand. Show your work!
x1 + 2x3 = 0
x1 + x2 + x3 = 0
2x1 − x2 + x3 = 1
1
40 pts.
Problem 4. In each part, determine if the given vectors span R3 .
A.

−1



1



4

2


6
−3




25
v4 = 


−23


,
−5
v3 = 


6




v2 = 
 3 ,
2



v1 = 
 1 ,
0
40 pts.



,
5
v2 = 


−4


,
5
v1 = 


−5
B.
0


12




v4 = 
 9 
7



v3 = 
 5 ,
2
Problem 5. In each part, determine if the given vectors in R4 are linearly
independent. If they are dependent, find constants c1 , c2 and c3 ,
not all zero, so that
c1 v1 + c2 v2 + c3 v3 = 0.
A.

−4



 4 


v1 = 
,
 −1 


1
B.

0



 1 


v1 = 
,
 2 


0

3



 1 


v2 = 
,
 −1 


0

4



 7 


v2 = 
,
 13 


21
2

10



 −2 


v3 = 

 −1 


−1

5



 4 


v3 = 

 3 


26
40 pts.
Problem 6. Let S be the subspace of R4 spanned





−3
0
1





 −33
 0 
 11 





v1 = 
 , v3 = 
 , v2 = 
 −26
 −2 
 6 





−2
1
2
by the vectors





,


0



 0 


v4 = 
.
 −1 


0
Consider the vectors

2


1



 11 


w2 = 
.
 −2 


5


 21 


w1 = 
,
 11 


3
Determine if w1 and w2 are in the subspace S. If so, write that vector as a
linear combination of v1 , v2 , v3 and v4 , i.e., If wj ∈ S, find constants c1 , c2 , c3
and c4 so that
wj = c1 v1 + c2 v2 + c3 v3 + c4 v4 .
40 pts.
Problem 7. Let S be the subspace of R4 spanned by the vectors







−1
14
3
19







 5 
 21 
 −2
 18 







v1 = 
 , v4 = 
 , v2 = 
 , v3 = 
 −2 
 0 
 1
 −3 







−1
22
4
32




.


These vectors are linearly dependent. Throw out vectors from this list until
you get an independent list, i.e., a basis for S. What is the dimenstion of S?
Justify your steps in the problem.
3