linear Algebra Problems Exam Practice

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Name:
Linear Algebra
Date:
Exam 2 – Take Home
Submission Protocols The completed quiz should be uploaded – as a single PDF file – to the MyMathLab Document Sharing forum. Please see the submission protocols sent to you via email. However, I summarize the document
naming protocols below:
Please put your “last name”, “first name”,“quiz number”,“submission number”,“course” in the file name –
separated by underscores (without the quotes or commas). For example, the filename for the first submission
for student John Doe should be: Doe John Exam2 P2 V1 2318 .
Quality of Submitted Work
Before you submit your quiz, you must first examine the PDF file to make sure that your work is legible and that there are no
large black borders surrounding the images (this will waste too much toner when printing). In other words, make sure that
your work is legible in the scanned image AND please try to ensure that I will not be wasting printer toner when printing.
Instructions Show all work and pertinent calculations (particularly when performing elementary row operations). An answer is not considered to be correct without complete and correct supporting work. In other words, for each
question, to receive full credit you must show all work. Explain your reasoning fully and clearly. I expect your solutions
to be well-written, neat, and organized. Do not turn in rough drafts. What you turn in should be the “polished” version
of potentially several drafts. Once again, show all of your work and justify your solutions fully. In addition, the simple
operational rules for this quiz are:
A. Calculators or computer software solutions will not be accepted.
Students are expected to present their own analytical solutions.
B. You may refer to theorems in the book (unless the question specifically states
otherwise), but only cite theorems that are from the appropriate sections in our
textbook; i.e., those that come from the relevant sections of Chapters 1, 2 and 3.
Nota Bene:
To receive full credit on graded work, it is important that you write each step of the row reduction process correctly. Furthermore, when completing a proof or justifying your work it is important that you explicitly state your conclusion and reasons.
When performing elementary row operations, please use a style that is similar to that outlined in the example.
Example. Show the that the linear system associated with
Solution.




1
2 −1
4
7
2
6 −7
 −2R1 + R2 → R2
1R1 + R3 → R3
−1 −1 −2 −1
∼


4 1R3 + R1 → R1
1 2 −1
3 
1 −3
 3R3 + R2 → R2
0 0
1 −7

 0
∼
∼



1 2 −1
4
2 −5 −1 

3 R3 ↔ R2
0 1 −3

1 2 0 −3 −2R2 + R1 → R1
1 0 −18 

0 0 1 −7

 0

 0

∼




4
1
2 −1
2
6 −7
7

−1 −1 −2 −1
is consistent.

1 2 −1
4
3 
1 −3

0 2 −5 −1 −2R2 + R3 → R3

 0
∼


33
1 0 0
1 0 −18 

0 0 1 −7

 0
.
Since the there is no pivot position in the rightmost column, we know the associated linear system is
consistent. Finally, the solution to the system is (33, −18, −7).
This portion (i.e., Part 2) of the entire quiz is worth 40 points.
Do not submit this cover page.
–Created/Revised by Mr. Sever 3/5/20
Page 0 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home
1. (6 pts.) Let A =
a) (A + B)2
b) (AB T )






1 2 2
−2
1 −1

3 1 2 
5
−3
4 
,
B
=


. Compute:
−1 2 4
−2
1
6
T
c) A2 − 3A + 4I
–Created/Revised by Mr. Sever 3/5/20
Page 1 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home
2. (8 pts.) Let A =




1 1 1
1 2 3 
. Write A as the product of elementary matrices.
1 4 5
–Created/Revised by Mr. Sever 3/5/20
Page 2 of 9
Linear Algebra
Name:
Date:
Exam 2 – Take Home
3. (9 pts.) Use the row reduction algorithm to find the inverse of A =
–Created/Revised by Mr. Sever 3/5/20





1
1
2
1
1 1
5
1 2
0
3 −1 1
0 1
2



.

Page 3 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home
4. (4 pts.) Determine which of the following matrices are elementary.




0 1 0
E1 = 1 0 0 ,
0 0 1
0 0 1
E7 = 0 1 0 ,
1 0 0
E2 =
”

#
1 −1
,
0
1

1 0 0
E8 = 0 0 0 ,
0 0 1


1 0 0
E3 = 0 2 0 ,
0 0 1
E9 =
”
√
#
π 0
,
0 1
E4 =
E10
#
”
0 0
,
0 0
”
#
0 1
,
=
1 0


1 0 −2
E5 = 0 1 0 ,
0 0
1
E11
”
#
0 2
,
=
1 0
E12
E6 =
”
#
1 0
,
5 1


1 0 0
= 0 1 0  .
0 0 1
In the list below, circle the matrices that are elementary:
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
E11
E12
5. (6 pts.) Construct a 4 × 4 matrix A and a vector b such that b is not in Col(A) and rank(A) = 2.
Show that your matrix has the desired property.
–Created/Revised by Mr. Sever 3/5/20
Page 4 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home
6. (6 pts.) Suppose that A, B, and X are n × n matrices with A invertible.

Solve XAT
T
+ 3AB 2 = AIn A for A.
7. (4 pts.) Given that det(AT ) =
8
for a 3 × 3 matrix A. Find the determinant of A.
5


1 4
3

8. (6 pts.) Find the determinant of A =  2 1 2 
.
4 3 −2
–Created/Revised by Mr. Sever 3/5/20
Page 5 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home
9. (10 pts.) The reduced echelon form of A =





3
0
4
1 −2 −5
−1
0
4
1 −2 −1
1
0
7
 0
2 −2 


 is 
 0
0
3 
0
2
2


0
1
0
0
0
0
1
0 −1 −1 

.
1
0
1 
0
0
0

a) Find a basis for the null space of A.
b) Find a basis for the column space of A.
c) State the dimension of the null space of A. Answer:
d) State the dimension of the column space of A. Answer:
–Created/Revised by Mr. Sever 3/5/20
Page 6 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home
10. (4 pts.) Let B =
11. (8 pts.) Let B =
(”
−1
−1







−3
7

1 
, 5
−4
−6
# ”
,
−2
1
#)
 
2
be an ordered basis of R . Find x ∈ R such that [x]B =








”
#
6
.
4

−3
be an ordered basis of H = Span{B}. Find  1  .
−4 B
12. (4 pts.) Find all values of h such that A =-
–Created/Revised by Mr. Sever 3/5/20
2
”
2h 4h
2 h2
#
is singular.
Page 7 of 9
Linear Algebra
Name:
Date:
Exam 2 – Take Home


13. (12 pts.) Use row reduction to find the determinant of A where A = 


–Created/Revised by Mr. Sever 3/5/20
1
2
3
1
4
1
2
0
−3
1
−3 −1 

.
−3
1 
−1
1

Page 8 of 9
Name:
Linear Algebra
Date:
Exam 2 – Take Home











a
b 
14. (13 pts.) Let H =
:a+b+c=0 .


c
a) Show that H is a subspace of R3 by finding a matrix A such that Nul(A) = H.
b) Using your result in part a), find a basis for H.
–Created/Revised by Mr. Sever 3/5/20
Page 9 of 9

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