Math 1553 Exam 3, SOLUTIONS, Spring 2025
Circle your instructor and lecture below. Be sure to circle the correct choice!
Jankowski (A+HP, 8:25-9:15 AM) Jankowski (C, 9:30-10:20 AM)
Al Ahmadieh (I, 2:00-2:50 PM) Al Ahmadieh (M, 3:30-4:20 PM)
Please read the following instructions carefully.
• Write your initials at the top of each page. The maximum score on this exam is 70
points, and you have 75 minutes to complete it. Each problem is worth 10 points.
• There are no calculators or aids of any kind (notes, text, etc.) allowed. Unless stated
otherwise, the entries of all matrices on the exam are real numbers.
• As always, RREF means “reduced row echelon form.” The “zero vector” in Rn is the
vector in Rn whose entries are all zero.
• On free response problems, show your work, unless instructed otherwise. A correct
answer without appropriate work may receive little or no credit!
• We will hand out loose scrap paper, but it will not be graded under any circum-
stances. All answers and work must be written on the exam itself, with no exceptions.
• This exam is double-sided. You should have more than enough space to do every
problem on the exam, but if you need extra space, you may use the back side of the
very last page of the exam. If you do this, you must clearly indicate it.
• You may cite any theorem proved in class or in the sections we covered in the text.
• For questions with bubbles, either fill in the bubble completely or leave it blank. Do
not mark any bubble with “X” or “/” or any such intermediate marking. Anything
other than a blank or filled bubble may result in a 0 on the problem, and regrade
requests may be rejected without consideration.
I, the undersigned, hereby affirm that I will not share the contents of this exam with
anyone. Furthermore, I have not received inappropriate assistance in the midst of nor prior
to taking this exam. I will not discuss this exam with anyone in any form until after 7:45
PM on Wednesday, April 9.
,This page was intentionally left blank.
, 1. TRUE or FALSE. Clearly fill in the bubble for your answer. If the statement is ever
false, fill in the bubble for False. You do not need to show any work, and there is no
partial credit. Each question is worth 2 points.
(a) If A is a 3 × 3 matrix, then det(2A) = 8 det(A).
True
⃝ False
1 0
(b) If A is a 3 × 3 matrix and A −1 = 0, then det(A) = 0.
1 0
True
⃝ False
(c) If A is a 3 × 3 matrix and
1 4
A 1 = 4 ,
3 12
then the equation Ax = 4x must have infinitely many solutions.
True
⃝ False
(d) If u and v are eigenvectors of an n × n matrix A, then u + v must also be an
eigenvector of A.
⃝ True
False
(e) If A is a 3 × 3 matrix with eigenvalues λ1 = 0, λ2 = 1, and λ3 = −4, then A must
be diagonalizable.
True
⃝ False
Circle your instructor and lecture below. Be sure to circle the correct choice!
Jankowski (A+HP, 8:25-9:15 AM) Jankowski (C, 9:30-10:20 AM)
Al Ahmadieh (I, 2:00-2:50 PM) Al Ahmadieh (M, 3:30-4:20 PM)
Please read the following instructions carefully.
• Write your initials at the top of each page. The maximum score on this exam is 70
points, and you have 75 minutes to complete it. Each problem is worth 10 points.
• There are no calculators or aids of any kind (notes, text, etc.) allowed. Unless stated
otherwise, the entries of all matrices on the exam are real numbers.
• As always, RREF means “reduced row echelon form.” The “zero vector” in Rn is the
vector in Rn whose entries are all zero.
• On free response problems, show your work, unless instructed otherwise. A correct
answer without appropriate work may receive little or no credit!
• We will hand out loose scrap paper, but it will not be graded under any circum-
stances. All answers and work must be written on the exam itself, with no exceptions.
• This exam is double-sided. You should have more than enough space to do every
problem on the exam, but if you need extra space, you may use the back side of the
very last page of the exam. If you do this, you must clearly indicate it.
• You may cite any theorem proved in class or in the sections we covered in the text.
• For questions with bubbles, either fill in the bubble completely or leave it blank. Do
not mark any bubble with “X” or “/” or any such intermediate marking. Anything
other than a blank or filled bubble may result in a 0 on the problem, and regrade
requests may be rejected without consideration.
I, the undersigned, hereby affirm that I will not share the contents of this exam with
anyone. Furthermore, I have not received inappropriate assistance in the midst of nor prior
to taking this exam. I will not discuss this exam with anyone in any form until after 7:45
PM on Wednesday, April 9.
,This page was intentionally left blank.
, 1. TRUE or FALSE. Clearly fill in the bubble for your answer. If the statement is ever
false, fill in the bubble for False. You do not need to show any work, and there is no
partial credit. Each question is worth 2 points.
(a) If A is a 3 × 3 matrix, then det(2A) = 8 det(A).
True
⃝ False
1 0
(b) If A is a 3 × 3 matrix and A −1 = 0, then det(A) = 0.
1 0
True
⃝ False
(c) If A is a 3 × 3 matrix and
1 4
A 1 = 4 ,
3 12
then the equation Ax = 4x must have infinitely many solutions.
True
⃝ False
(d) If u and v are eigenvectors of an n × n matrix A, then u + v must also be an
eigenvector of A.
⃝ True
False
(e) If A is a 3 × 3 matrix with eigenvalues λ1 = 0, λ2 = 1, and λ3 = −4, then A must
be diagonalizable.
True
⃝ False
