additional sample1 and solutions (1)

8/19/2019 Additional Sample1 and Solutions (1)

1/15

Math 1B03

1st Sample Test #1

Name:___________________________________________

(Last Name) (First Name)

Student Number: Tutorial Number: _____________________ ____________________

This test consists of 20 multiple choice questions worth 1 mark each (no part marks), and 1

question worth 1 mark (no part marks) on proper computer card filling. All questions must be

answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for

wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your

copy of the test is complete. Bring any discrepancy to the attention of the invigilator.

Calculators are NOT allowed.

1. Which of the following matrices are in reduced row echelon form?

(i) (ii) (iii)

" # ! $ !! ! " " ! " ! $ "! ! ! ! " ! " # %! ! ! ! !

" ! ! &! ! " $! " ! %

(iv) " ( & &! " $ #(a) (i), (iii), and (iv) only

(b) (iii) only

(c) (i), (ii), and (iii) only

(d) (i) and (iii) only

(e) none of them

2. Let . Find the reduced row echelon form of .E œ E# " " !" $ & "$ " ( #

(a) (b) (c)

" ! ! " ! ! " ! !

! " ! ! " ! ! " !

! ! ! " ! ! ! " ! ! ! "

) ) )& & &

* "" ""& & &

(d) (e)

" ! !

! " !

! ! ! "

" ! !

! " !

! ! ! "

)&

""&

(&

""&


2/15

3. Suppose that the augmented matrix for a system of linear equations has been reduced by row

operations to the given reduced row echelon form. Solve the system.

" ' ! ! $ #! ! " ! % (

! ! ! " & )! ! ! ! ! !

(a) (b) (c) B œ #= $> '? B œ # $= '> B œ # $> '=B œ ? B œ > B œ =B œ (= %> B œ ( %> B œ ( %>B œ )= &> B œ ) &> B œ ) &>B œ > B œ = B œ >

" " "

# # #

$ $ $

% % %

& & &

(d) (e) B œ # $> ' B œ # $> '=B œ ! B œ =B œ ( %> B œ ( %>B œ ) &> B œ ) &>B œ > B œ >

" "

# #

$ $

% %

& &

4. Solve the following system of equations

#B B B B #B œ "

$B $B #B $B œ !

$B B $B "#B œ "

" # $ % &

" # $ &

# $ % &

(a) (b) (c)no solution B œ $ #= %> B œ $ #= )>

B œ " #> B œ =B œ & $= > B œ ' $= "&>

B œ = B œ " #= "#>

B œ > B œ >

" "

# #

$ $

% %

& &

(d) (e)B œ $ #= )> B œ " = %>

B œ " > B œ " $= >

B œ ' $= "&> B œ =

B œ = B œ #= '>

B œ > B œ >

" "

# #

$ $

% %

& &

5. If can be formed, is , and is , what size is ?EFG E $ ‚ # G % ‚ & FX

(a) (b) (c) (d) (e) # ‚ # # ‚ & $ ‚ % # ‚ % $ ‚ &


3/15

6. Find conditions on and such that the following system has exactly one solution+ ,

B ,C œ "

#+B #C œ &

(a) and+, œ " + Á (b) +, Á "(c) + œ ß , œ & ## &(d) + œ $,ß , Á #&(e) +, œ #ß + Á &

#

7. Consider the following system.

#B C #D œ &

B C $D œ "

B #C %D œ '

Given that the inverse of is equal to , which of the

# " #" " $" # %

"! ) ""$ "$ "$" ' %

"$ "$ "$$ & "

"$ "$ "$

following gives a solution to the above system?

(a) (b)

"! " $ "! ) ""$ "$ "$ "$ "$ "$

) ' &"$ "$ "$

" % " $ & ""$ "$ "$ "$ "$ "$

" ' %"$ "$ "$

" "& &

' '

(c) (d) "! " $ "! ) ""$ "$ "$ "$ "$ "$

) ' &"$ "$ "$

" % " $ & ""$ "$ "$ "$ "$ "$

" "

" ' %"$ "$ "$

" "

& &

' '

(e) none of the above

8. Find the matrix if E

ÐE #M Ñ œ $ "

# #X "

(a) (b) (c) E œ E œ E œ! # " #

" " " &

& "# #

" ""% %

"%

(d) (e) E œ E œ" #

" &

$ "# #" &

% %


4/15

9. Find an elementary matrix such that .I F œ IE

E œ ß F œ$ ' " ## % # %

(a) (b) (c) (d) (e) " " " # " " " # " ## # # # ! " ! " ! "10. Which of the following matrices are symmetric.always

(i) (ii) (iii) for any scalarE E EE 5E 5X X (iv) E EX

(a) (i), (ii), and (iii) only

(b) (i), (ii), and (iv) only

(c) (ii) and (iv) only

(d) (i) and (ii) only

(e) (i), (ii), (iii), and (iv)

11. Given that det , compute det .

< = > < = >? @ A B )< C )= D )>B C D )? )@ )A

œ %

(a) (b) (c) (d) (e) $# $# #&' #&' !

12. If , calculate .det det + ,- . œ $# # !

- " " #+. # # #,

(a) (b) (c) (d) (e) % "# "# % $

13. If is and , find .E $ ‚ $ Ð#E Ñ œ $ œ ÐE ÐF Ñ Ñ Fdet det det" $ " X

(a) (b) (c) (d) (e) $ ) ) # #) $ $ $ $# $ $ $ $

$ % # % #

14. Compute the determinant of the following matrix,

$ " & #" $ ! "" ! & #" " # "

(a) (b) (c) (d) (e)$" "$# "$" "$! !


5/15

15. Find the adjoint of the following matrix .

" " #$ " !! " "

(a) (b) (c) " " % " " % " " #

* " ! * " ! $ " '! " % ! " % $ " %

(d) (e)

" " # " $ !$ " ' " " "$ " % # ! "

16. Find the eigenvalues of .E œ# ! !" # "" $ #

(a) (b) (c) (d) (e) #ß "ß " "ß " #ß " #ß " #ß "ß !

17. Suppose that a matrix (not given) has eigenvalues with eigenvectorsE œ "ß #ß $-

" " !# ! !" " "

ß ß T H T ET œ Hand , respectively. Find and so that ."

(a) ,T œ H œ" # " " ! !

" ! " ! # !

! ! " ! ! $

(b) ,T œ H œ

! " " $ ! !! # ! ! " !" " " ! ! #

(c) ,T œ H œ

" " ! " ! !! # ! ! $ !" " " ! ! #

(d) ,T œ H œ" " ! $ ! !# ! ! ! # !" " " ! ! "

(e) ,T œ H œ" ! " " ! !# ! ! ! # !" " " ! ! $


6/15

18. Suppose for some diagonalizable matrix (not given). Calculate:Ð Ñ œ Ð "Ñ $ ‚ $ E- - $

E#&.

(a) (b) (c)

! ! ! #& ! ! #& ! !! ! ! ! #& ! ! #& !

! ! ! ! ! #& ! ! #&

(d) (e)

" ! ! " ! !! " ! ! " !! ! " ! ! "

19. xSuppose that is an eigenvalue of with eigenvector , and is an eigenvalue of with- -" #E Fthe same eigenvector . Consider the following statements.x

(i) is an eigenvalue of the matrix- -" # ÐE FÑ(ii) is an eigenvalue of the matrix- -" # FE(iii) is an eigenvalue of the matrix-"

$ $E

Which of the above statements are always true?

(a) (i), (ii), and (iii)

(b) (i) and (ii) only

(c) (i) and (iii) only

(d) (ii) only

(e) (i) only

20. In Matlab what command could be used to create the row vector

( ?$ß&ß(ß*ß""ß"$ß"&ß"(ß"*Ñ(a) (b) (c) >>[ 3 by 2 t o 19] >>3: 2: 19 >>[ 3 t o 19 by 2]

(d) >>f or ( i = 3 t o 19 by 2) x[ i ] = i end

(e) >>[ 3; 5; 7; 9; 11; 13; 15; 17; 19]


7/15

21. Correctly fill out the bubbles corresponding to your student number and the version number

of your test in the correct places on the computer card. (Use the below computer card for

this sample test.)


8/15

Math 1B03

2nd Sample Test #1

Name:___________________________________________

(Last Name) (First Name)

Student Number: Tutorial Number: _____________________ ____________________

This test consists of 20 multiple choice questions worth 1 mark each (no part marks), and 1

question worth 1 mark (no part marks) on proper computer card filling. All questions must be

answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for

wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your

copy of the test is complete. Bring any discrepancy to the attention of the invigilator.

Calculators are NOT allowed.

1. F and that express the given system of linear equations as a singleind matrices ,E \ Fmatrix equation .E\ œ F

%B $B B œ "

&B B )B œ $

#B &B *B B œ !

$B B (B œ #

" $ %

" # %

" # $ %

# $ %

(a) E œ \ œ F œ

! % $ " B "! & " ) B $

# & * " B !! $ " ( B #

"

#

$

%

(b) E œ \ œ F œ

% ! $ " " B !& " ! ) $ B !# & * " ! B !! $ " ( # B !

"

#

$

%

(c) E œ \ œ F œ

% ! $ " " B "& " ! ) $ B $# & * " ! B !! $ " ( # B #

"

#

$

%

(d) E œ \ œ F œ

% ! $ " B "& " ! ) B $# & * " B !! $ " ( B #

"#

$

%

(e) E œ \ œ F œ

! % $ " " B !! & " ) $ B !# & * " ! B !! $ " ( # B !

"

#

$

%


9/15

2. Consider the matrices

E œ G œ ß I œ$ ! ' " $

" # " " #" " % " $

" % #$ " &

,

Compute , if possible.G E #I X X X

(a) (b) (c) (d) undefined

"& ( "! "& $ "# "& "% "#"! ! * "% ! ( $ ! "#"% "! "$ "# "# "$ "# ( "$

(e)

"& "! "%( ! "!

"! * "$

3. Solve the following system of equations#B B B B #B œ "

$B $B #B $B œ !

#B B B B œ "

" # $ % &

" # $ &

" # $ %

(a) (b) (c)no solution B œ $ #= %> B œ $ #= )>

B œ " #> B œ =

B œ & $= > B œ ' $= "&>

B œ = B œ " #= "#>

B œ > B œ >

" "

# #

$ $

% %

& &(d) (e)B œ $ #= )> B œ " = %>

B œ " > B œ " $= >

B œ ' $= "&> B œ =

B œ = B œ #= '>

B œ > B œ >

" "

# #

$ $

% %

& &

4. Use determinants to find all of the possible real values of which make the following matrix+not invertible.

E œ" " +

+ " ++ " #

(a) (b) (c) (d) (e) and# " „ " " „ # !


10/15

5. Find conditions on , , and such that the system has infinitely many solutions+ , -

-B $C #D œ )

B D œ #

$B $C +D œ ,

(a) + - & Á !(b) and+ - œ ! , #- # œ &(c) and+ - & œ ! , #- # œ !(d) and+ - œ ! , #- # Á &(e) and+ - & œ ! , #- # Á !

6. Find the diagonal entries of the inverse of .

$ " #" " $

" # %

(a) (b) (c)

# # #& & &

# # #& & &

% % %#& #& #&

‡ ‡ ‡ ‡ ‡ ‡

‡ ‡ ‡ ‡ ‡ ‡

‡ ‡ ‡ ‡ ‡ ‡

(b) (b)

# #& &

# #& &

% %#& #&

‡ ‡ ‡ ‡

‡ ‡ ‡ ‡

‡ ‡ ‡ ‡

7. Consider the following matrix,

E œ " #

! $ Note that can be reduced to using the following row operations:E M

(i) r > r # #"$

(ii) r > r r " " # #

Using the above two row operations in the above order, find elementary matrices andI I " #

such that .E œ I I " #" "

(a) (b) I œ ß I œ I œ ß I œ" ! " !

! !" # " #! " ! "" # " #" "

$ $

(c) (d) I œ ß I œ I œ ß I œ" ! " # " #! $ ! " ! "

" !

!" # " # "$

(e) I œ ß I œ" !

!" #! "" #"

$


11/15

8. Suppose that and are symmetric matrices. Which of the following matrices areE F alwayssymmetric?

(i) (ii) (iii)E EF EF FE"

(a) (b) (c) (d)(i) only (i) and (ii) only (i) and (iii) only (ii) and (iii) only

(e) none of them

9. If , which of the following is equal to ?E œ ! ÐM EÑ$ "

(a) (b) (c) (d) (e) M E M E E M E M E E M E E# # #

10. skew-symmetricA matrix is if . Suppose that and are both skew-E E œ E E FX

symmetric. Which of the following matrices are always skew-symmetric?

(i) (ii) (iii)E F EF 5E

(a) (i) only

(b) (i) and (iii) only

(c) (iii) only

(d) (i), (ii), and (iii)

(e) (ii) only

11. Consider the following statements,

(i) for all matrices and .ÐE FÑ œ ÐF EÑ 8 ‚ 8 E F# #

(ii) det detÐE F Ñ œ ÐE FÑX X

(iii) If then or .EF œ ! E œ ! F œ !


(a) (i) only



(d) (ii) and (iii) only

(e) all of them

12. Let . Given that use the adjoint to find the entry inE œ E œ %ß

" # % "

% ! " #$ $ ( !$ & ' %

det

row 1 column 2 of .E"

(a) (b) (c) (d) (e) * * '&% % #

* *


12/15

13 idempotent. if . If is idempotent, which of theA square matrix is calledT T œ T T #

following matrices are also idempotent?

(i) (ii) (iii)MT M T M #T

(a) (i) only

(b) (i) and (ii)(c) (i) and (iii)

(d) (ii) only

(e) (i), (ii), and (iii)

14. If is and , find adj .E $ ‚ $ E œ # ÐE % EÑdet det "

(a) (b) (c) (d) (e) $'% $'& (#*(#* $'&# #

15. Let and assume that . Compute whereE œ E œ # Ð#F Ñ+ , -: ; <? @ A

det det "

F œ%? #+ : %@ #, ; %A #- <

.

(a) (b) (c) (d) (e)" "' # " "# %

16. Let and be matrices. Consider the following statements.E F 8 ‚ 8(i) det detÐEFÑ œ ÐFEÑ(ii) det det detÐE FÑ œ E F(iii) det detÐEÑ œ ÐEÑ


(a) (i) only



(d) (i), (ii), and (iii)

(e) (iii) only


13/15

17. Given that the matrix has as one of its eigenvalues, find theE œ œ "! " "" ! "" " !

-

corresponding eigenvector(s).

(a) (b) (c) (d) and and " " " " " !" ! " ! " "! " ! " " "

(e) and

" "" "" !

18. Find a matrix such that is diagonal.T T ET E œ! ! !! $ #! " !

"

(a) (b) (c)

" ! ! " ! ! " ! !! " # ! " # ! " #! " " ! " " ! " "

(d) (e)

" ! ! " ! !" " # ! " #! " " " " "

19. Consider the following statements.

(i) If is diagonal, and is diagonal, then diagonalizable.T ET T FT EF" "(ii) If is diagonalizable then , where are the (notE ÐEÑ œ â ß ß á ßdet - - - - - -" # 8 " # 8necessarily distinct) eigenvalues of .E(iii) If is diagonalizable then must be invertible.E E


(a) (ii) only

(b) (ii) and (iii) only

(c) all of them

(d) (i) and (iii) only

(e) (i) and (ii) only

20. xIn Matlab, suppose that we have defined a vector , and we want to square every component

of the vector . Which command could accomplish this?x

(a) (b) (c) >>x 2̂ >>square( x) >>x[ 1] 2̂, x[ 2] 2̂, . . . , x[ n] 2̂

(d) (d) >>x. 2̂ >>f or i = 1 t o si ze( x) x[ i ] = x[ i ] 2̂ endf or


14/15

21. Correctly fill out the bubbles corresponding to your student number and the version number

of your test in the correct places on the computer card. (Use the below computer card for

this sample test.)


15/15

Answers for 1st Sample Test #1

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.d c e a b b b b e d

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. b c b b d a b e a b

21.

NOTE: On the sample tests, a version number is not given. On the actual tests, it will say

"Version X" at the top, where X is the version number that you will have to fill in on thecomputer card. The sample answer above assumes that the test says "Version 3" at the top.

On the actual test you will have to fill in the bubble corresponding to the version number of

YOUR test (which may or may not be Version 3). The sample above also assumes that your

student number is 8816132. On the actual test, you will have to fill in the bubbles

corresponding to YOUR student number (not 8816132).

Answers for 2nd Sample Test #1

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.d b d a c a a a b b

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. b a a b b a a c e d 21. see the answer to #21 on the first sample test above.

additional sample1 and solutions (1)

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