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Math 214, Final Exam 2021

Question 1 (1 points)

As the LSA Community Standards of Academic Integrity says, “The LSA undergraduate academic com- munity, like all communities, functions best when its members treat one another with honesty, fairness, respect, and trust . . . the College promotes the assumption of personal responsibility and integrity and prohibits all forms of academic dishonesty and misconduct.  Academic dishonesty may be understood as any action or attempted action that may result in creating an unfair academic advantage for oneself or an unfair academic advantage or disadvantage for any other member or members of the academic community.”

On this exam, you may consult a three-page note card prepared in advance and an approved calculator, but you may not consult any other resource, or receive help from any other person.

For one point, please copy onto your exam and sign the statement“I affirm that I have not received help from any other person on this exam, nor consulted any inappropriate resource.”






Signature:


Question 2 (10 points)

Let A be a 4 × 4 matrix with eigenvalues 2021, 1, and 1/2021. Let v1  be an eigenvector with eigenvalue 2021, let v2 and v3 be two linearly independent eigenvectors with eigenvalue 1, and let v4 be an eigenvector with eigenvalue 1/2021.  If your answer is infinity write ∞ .  Show representative work outside the box and the answer inside the box.

(a)  (4 points) If ⃗v = 2⃗v2 + ⃗v3 + ⃗v4 , then compute A2021 ⃗v.

Answer


(b)





(c)


(4 points) If w= 3⃗v1 + ⃗v4 , then compute A2021 w⃗ .

Answer

(2 points) Determine whether the following statement is true or false. limn →∞ An ⃗v = limn →∞ An w⃗ .

Answer


Question 3 (12 points)

Let c be a real number and consider the real matrix

A = 9(c)   c(−)1 .

No explanation is required for your answer.


Answer

(b)  (6 points) Find an eigenvector corresponding to each eigenvalue.  Write ⃗vλ = ···  below with the

subscript λ replaced by the numerical value of each eigenvalue in part (a).

Answer


(c)  (2 points) Is this matrix A similar to a real matrix of the form b(a)  the box.   (Matrices A  and B  are similar if there is an invertible


a(−)b ? Answer yes or no inside

matrix S  with A  = SBS 1 .)


Question 4 (9 points)

For the following questions, write TRUE if the statement is true, and FALSE if the statement is false. No justification necessary.


(a)  (3 points) Suppose A is an n ×m matrix with n < m . Then AT A is not invertible.


(b)  (3 points) Suppose A  and B  are two n × n matrices which satisfy det(A) = −2 and det(B) = 9.

Then det(A + B) = 7.


(c)  (3 points) Let A and B  be two n × n matrices. If B  is obtained from A by exchanging two rows, then det(B) = det(A).


Question 5 (11 points)

Let A be 2021 × 2021 matrix such that each entry in the matrix is 2021.

(a)  (4 points) The rank of A  is...(write a number or  “UND” if the rank is UNDetermined from the

information given)                             .

(b)  (5 points) The real eigenvalues are, with their multiplicities  (list the eignevalues in decreasing

order):


Eigenvalue, λ

Algebraic multiplicity

Geometric multiplicity



(c)


(2 points) True or false: The matrix A is diagonalizable.



Question 6 (15 points)

Suppose A is a 3 × 3 matrix which satisfies det(A) = − 13. Write A in terms of column vectors as

A =  ⃗v1     ⃗v2     ⃗v3   .

(a)  (5 points) Define a matrix B by

B = ⃗v1 + ⃗v2 ⃗v1 ⃗v2 ⃗v3   .

Calculate det(B) or write “UND” if the value is UNDetermined by the information given.


det(B) =



(b)  (5 points) Define a matrix C by

C =  2⃗v1 − 2⃗v3      2⃗v2 + 4⃗v3 ⃗v1 + ⃗v2 + v3   .

Calculate det(C) or write “UND” if the value is UNDetermined by the information given.


det(C) =



(c)  (5 points) Define a matrix D by

D =   − 3v3 v1 v2   .

Calculate det(D) or write “UND” if the value is UNDetermined by the information given.


det(D) =



Question 7 (10 points)

(a)  (5 points) Let B denote a 2 × 2 matrix, and suppose multiplication by B transforms the shape on

the left to the shape on the right:


y


(−1

, 1)

,


Suppose that we know B 1 =


y


,

,

2(2) . Calculate det(B).


(b)  (5 points) Let C denote a 2 × 2 matrix, and suppose multiplication by C transforms the shape on

the left to the shape on the right:



y


(1,

1)

(4,

0)



C


y