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Math 201: Linear Algebra Name (Print):

Spring 2020

Exam 2 JHU-ID:

4/20/20

Time Limit: 50 Minutes Section Number ( & Time)

This exam contains 9 pages (including this cover page) and 8 problems. Check to see if any pages

are missing. Enter all requested information on the top of this page, and put your initials on the

top of every page, in case the pages become separated.

You may not use your books, notes, or any calculator on this exam.

You are required to show your work on each problem on this exam. The following rules apply:

• If you use a “fundamental theorem” you

must indicate this and explain why the theorem

may be applied.

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coherent way, in the space provided. Work scat-

tered all over the page without a clear ordering

will receive very little credit.

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receive full credit. A correct answer, unsup-

ported by calculations, explanation, or algebraic

work will receive no credit; an incorrect answer

supported by substantially correct calculations and

explanations might still receive partial credit.

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clearly indicate when you have done this.

Do not write in the table to the right.

Solutions

Math 201: Linear Algebra Exam 2 - of 9 4/20/20

1. (a) (12 points) Let

A :=

2

66664

6 0 6 0 1

0 1 0 3 0

1 0 6 0 2

0 0 0 1 0

6 0 2 0 2

3

77775

Calculate Det(A).

(Double check your calculation: your answer should be a whole number whose unit digit

equals either 0 or 2 or 4. Your answer could be a one digit number).

(b) (2 points) Can the matrix A be row reduced into its reduced row-echelon form without

using division?

Det'D

µ¥,±⇐#,i::÷÷÷÷⇒÷÷÷÷.④ Haut - Iv

Iu.ua#foioio GD

to:* .¥¥f⇐→L%¥÷:] 9

EH•-HI Track how

-D= 4. ( -18.5) determinant

D=p①Chases .

⑥ Why'd If you could row reduce w o

usingdivision A-

9- (which is

calculated we now reduction) would

have integer entries .But this can't

occur as det ( A- Y = May , is not an integer.

Joel Benjamin Specter


2. (15 points) Let A =

�8 �4

2 1

�

FIRST

Find all vectors v so that the distance between Av and the standard unit basis vector e1 is

minimized. Call the set of all such vectors L.

THEN

Find the unique vector v0 in L such that v0 is orthogonal to the kernel of A.

What is the x-coordinate of the vector v0?

(Computational Check: The sum of the numerator and denominator of the x-coordinate of v0is 77).

(Computational Check: It may be helpful to draw L and the orthogonal complement of the

kernel of A on a coordinate plane, and see that their intersection agrees with the vector v0which you’ve computed.).

we are tying to find approximate solutions to Av-

-

eye-

q t

To do this: solve Norad equation A' A - = Aea .

① ca¥ ②nowuedneecrata.EE#-I--4tJ' Catalina -- Gh 't;9¥¥i

⇒ E: .at:D68 344 I ⇒ EE 1-41." ) .

⇐ [34 Mt] '

solution L -- { It] - C. ii.d) *R}Ii÷÷÷i÷of

'

. ÷÷¥÷÷ . ,.\ be orthogonal to basis i.e . [ L?

Ruffa)= To ! ] "

I t.dgs.in that :

Kenta)-_Golf 'd ) ,←qq.pe#efo.y..J) , so -8kt

line of slope -2 I ~ Sse : 8/88

( through 0 ) in!÷¥fE)tE*-coordwate=•8/zs-

- 8-185=77¥:*.


3. Let A :=

19 �9

18 �8

�,

(a) (10 points) An eigenvalue of A is �1 := 1. What is the other eigenvalue of A?

(b) (11 points) Find an Eigenbasis for A. What are the slopes of your eigenvectors?

(The slope of a vector

xy

�in R2

isyx . Double check that you did not accidentally answer

incorrectly by writing the reciprocal.)

(Computational Check: The sums of the numerator and denominator of each of these

fate) -- (M - E) C - 8 - t) - 18C- 9)

= E - ut e @ 9h08 - 181-97]I E - H t t [at-8) - 18 C- 8) to 18 ]

= I - ht t 10.I

fate)= It - DH - ta) -- E - Clerk)-L

so

Ea..-

- Kelli: ::)) -

- Ker ( too)) -- spank :DS

Ea..

= he II. Id) -- rel co- '

o )) -- spell 'd)

Sl


slopes should be 3 and 2, respectively).


(c) (4 points) Consider the vector v :=

�75308644

�175308644

�.

How many trailing zeros appear at the end of the x-coordinate of A2020v? (A trailing zero

of an integer is 0-digit to the left of the decimal point before which any non-zero digit has

occurred, e.g. 312,000 has 3 trailing 0’s and 28,000,000 has 6 trailing 0’s.)

r-

- al : ) + a s -- Li :],

s"

-

- E. - i]

a:*:-p:÷:÷:". :÷÷: Em..

2020

A'"

:-. 10%1: ) - ask.lt)i

- O . .

€.÷:÷ . d= [ - * O O B 000000-00]

-⇒ . .

. O O Z o O O o yoo O O

then-U


4. (2 points) Is the following matrix diagonalizable?

2

66666666664

1 0 0 0 0 0 0 0

8 5 0 0 0 0 0 0

3 7 9 0 0 0 0 0

5 1 3 2 0 0 0 0

7 8 3 2 3 0 0 0

4 2 3 1 4 4 0 0

1 1 1 1 1 6 0 0

4 5 3 2 1 2 4 6

3

77777777775

.i.Since thus not- in

its eigenvalues ane lis ,9, 2,3 ,40,6

(the diagonal entries)counted

with algebraicmultiplicity .

As there are8 dstcf eggnukes

this 8×8 retryis diagonal#He .

( see video (2) on 4/13)


5. (4 points) Determine which properties below are always true for an n⇥ n matrix A:

1. det(A) = det(AT)

2. (A+ cI)T = AT+ cI for all real numbers c.

3. The characteristic polynomials of A and ATare equal.

4. The matrices A and AThave the same set of eigenvalues, and for each eigenvalue � the

geometric multiplicity of � as an eigenvalue of A equals the geometric multiplicity of � as

an eigenvalue of AT .

5. The matrix A2is invertible if and only if AT

is invertible.

AHT@d) True : ( see videos from 4/13)..f(2) True

-

(AtcI)T= Ate@IT= AT t cIa--e------.________.--#

(3)True:This as fact is applied to

A-ttifo-ec.lt#(4) True (

At -II) and LAHIJ-- AIHI

Lane

"

the same rank (see

video 3 for 4/13) .

-

(5) True both are equivalent to'

.

rank (A) ='t


6. (2 points) Let T : R2 ! R

2be the linear transformation that encodes rotation by 180 degrees.

Does T have an eigenbasis?

7. (2 points) How many n⇥n matrices have a single eigenvalue of 1 with a geometric multiplicity

of n?

¥. i÷*¥The matrix -s (I ?) which is dig!So e

,and ez are an eigen

bases

(IN feet > any basis of R2 is an eagerbasis)since Rot = -X

There is I such nah.x : the ident.ly'

since

TA -- S I. I) s- '= S5' -- I

See video 2 : From 4/13 .

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