Name _________________________
Exam II-A Makeup
Score ___________________
Section I. Short Answer Problems. Answer the problems on separate paper. You do not need to rewrite the
problem statements on your answer sheets. Work carefully. Do your own work. Show all relevant
supporting steps! Attach this sheet to the front of your answers.
1. (5 pts)
Find the determinant of the matrix A using an expansion by co-factors
1 2 0 3

2 1 1 2

where A 
 3 2 1 0 


 0 1 1 2
2. (5 pts)
Find the values of x for which the determinant of the matrix A is zero
1 x
0 1


0 1 x 2 

where A  
x 0 1 2


1 1 2 0 
3. (5 pts)
Find the determinant of the matrix A by using elementary row operations to reduce A to a
 2 1 2
3


2 1 0 1

triangular matrix A  
3 2 1 0 


1 2 6 
 0
4. (5 pts)
Give an example to show that the determinant of a sum of matrices is, in general, not equal to
the sum of the determinants of the matrices.
5. (5 pts)
 1

2
Determine whether the matrix A is singular where A  
1

 2
6. (6 pts)
 2 1 0
 1 4 1

A

Let
 3 1 2

0 2 1
7. (6 pts)
Let u = (2,2,-1,1), v = (-1,-3,-2,1), w = (2,-1,2,3).
Find: (a)
2 0
3 1
1 2
0 2
2
2 
T
. Find the determinant of: (a) A
1

2
3u - 4v - 6w (b) 2v + 3u - 4w
(b)
3

0
4

1 
A5
(c)
6A
Section II. Answer these problems on these test pages.
8. (3 pts)
A.
For each of the following sets answer the question: Is this set with its specified operations a
vector space? If the answer is no, give a reason for why the answer is no.
S =  p( x)  P | p (1)   p (1) ,
addition is standard polynomial addition,
scalar multiplication is standard scalar multiplication for polynomials
B.
S = (a, b) | a, b  0 ,
addition is defined by (a, b)  (c, d )  (ac, bd )
scalar multiplication is defined by  ( a, b)  (e a, e b)
C.
A.
B.
C.
9. (6 pts)
a 0 0 



S =  0 b 0  | a, b    ,


  0 0 ab 




addition is standard matrix addition,
scalar multiplication is standard scalar multiplication
Yes
Yes
Yes
No
No
No
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For each of the following sets answer the question: Is this subset of P3 a subspace of P3? If
the answer is no, give a reason for why the answer is no.
A. { p(t)  P3 | p(t) * p(-t) = 0 for all t }
B. { p(t)  P3 |



p(t)dt = 0 }
C. { p(t)  P3 | p(1) *p(-1) = 0 }
D. { p(t)  P3 | p(1) *p(-1) = 1 }
E. { p(t)  P3 | p(0) = p(1)=p(2) }
F. { p(t)  P3 | p'(t) = t* p''(t) }
A.
B.
C.
D.
E.
F.
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
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10. (6 pts)
For each of the following sets answer the question: Is this set a spanning set for the given
vector space?
A. { (3,5), (-6,–4) } for
2
Yes
No
Yes
No
Yes
No
E. {x  1, x  1, x  1}, P2
Yes
Yes
No
No
F. {x  1,  x 2  x  1, x 2  x  1}, P2
Yes
No
B. { (1,-2), (-1,2), (-5,2) },

2

3
D. { (-2,1,2), (2,-1,1), (2,-1,3) }, 
C. { (-2,-1,-1), (2,5,-3), (-2,-4,2) },
2
11. (6 pts)
3
2
For each of the following sets answer the question: Is this set a linearly independent subset of
the given vector space?
A. { (3,5), (-6,–4) } for
2
Yes
No
Yes
No
Yes
No
E. {x  1, x  1, x  1}, P2
Yes
Yes
No
No
F. {x  1,  x 2  x  1, x 2  x  1}, P2
Yes
No
B. { (1,-2), (-1,2), (-5,2) },

2

3
D. { (-2,1,2), (2,-1,1), (2,-1,3) }, 
C. { (-2,-1,-1), (2,5,-3), (-2,-4,2) },
2
12. (6 pts)
3
2
For each of the following sets answer the question: Is this set a basis for the given vector
space? If the answer is no, give a reason for why the answer is no.
A. { (3,5), (-6,–4) } for
2
B. { (1,-2), (-1,2), (-5,2) },
2
3
3
D. { (-2,1,2), (2,-1,1), (2,-1,3) }, 
C. { (-2,-1,-1), (2,5,-3), (-2,-4,2) },
E. {x  1, x 2  1, x 2  1}, P2
F. {x  1,  x 2  x  1, x 2  x  1}, P2
A.
B.
C.
D.
E.
F.
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
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