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Quiz 01 - Solving Systems of Linear Equations through Row Operations, Exercises of Linear Algebra

The solutions to quiz 01 of math 205b, focusing on finding the reduced row echelon form (rref) of augmented matrices and solving systems of linear equations using elementary row operations. Students are required to find the rref of three matrices (a, b, and c) and use the results to find the solutions of the corresponding systems.

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

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Math 205B Quiz 01 page 1 01/15/2010 Name~'t *~ r./fip.
1. Considerthe augmented matrix A= [1~ -i 11~~].
1A. What system of linear equations is represented by A?
{/h'X I- I.-/xz.~/OD
1.,., .,...><1.- ZL(
lB. Find the reduced row echelon form (rref) of A by hand, showing all the matrires produced along the way. Say what
each elementary row operation was used to produce each new matrix from the previous one using the "rs +- r3 +4r2" style
notation we developed in class)
~At< I!J&Yf..JJ. .fI!~ j"""1 .a ,,*,:.1.""'" led ,b,f iho</Y''l/;J. lIdl'i// -:
[
I, ''1
/
'OO
J~['-I -I 7-S'"
J~[I-2. /'Jr-- [I-2.
I
'J
3I2'( 3 f 2.'1 3Il&j r.~r.-'r. 0?21
1. t),
;:~rLlr [
t-l..
I
17 r,~ t; ...t,1. [,DI71
r--Y 0 I 3J ~ DI3J
1C. What is the solution of the system in part 1A?
[
'I. =;t
x\ =3 '
QdJ
z. 1 3 -3 11 11
2. Consider the system of equations whose augmented matrix is B= rrD 2 5 12 8 ~.
~~)j 1 13 13 11 ~
Use your calculator to find rref(B) and write the result in the space below. Circlethe pivot columns of B. Then use the
rref to findallthe solutionsof the system usingthe methods and notation we'vedevelopedin class. If there are no solutions,
explain why. Otherwise, givea specificsolution and show it satisfies the first equation the augmented matrix represents.
/,~ [I0J 2.
/
O
J7L kt ~r~ Ii ~v~
rrt{( IS/ =~ ~ -~ ~ f OxIrDx. ~0.I ~ Clx, "" 1),A,d. ~!!ls/cf1i,,~.
l' of /"",,11M
L\ p'-vet.cJt ~~ nil. M ~~syd:1Ylr~' 11; ~UK
(su. g) ,J. . "" .L.
~fk. fjsf£,(YI f(J IflCrfrt,Id~t...
[J
1
0
-3
W
11
]
3. Now consider the system of equations whose augmented matrix is C=:3 2 5 2 8 .
:13 1 11
Again, use your calculator to find rref( C) and write the result in the space below. Circle the pivot columns of C. Finally,
use the rref to find all the solutions of the system using the methods and notation we've developed in class. Hthere are
no solutions, explain why; otherwise, give a specific solution and show it satisfies the first equation the augmented matrix
represents.
[IJ
2 3
(0302. )( -= -X
Jkrt
)rrt{(C)::: D l -l. () -'" ~Wt.fa ,,' -l.i 2.
1
l'() () 0, .3 ."z-I+XJ
\ \ I)<.1 ~f;.u.
FWJvMII$~l. c.) )<~ == -)
Skt.)( It, ~IL.~oQ-mart" .rJ~{. 74/u Xj:: 0t/W ~!jl (2) 'fIOJ-j)~
1,/ ,p .
tJMsPtc,ihl.sJJW;.. l1k /11vIfckJr It in fk tI~VMl;o XI+-JX1.-1~ J + ~'1 =II :
f7 7 /'
2i-3(f4) -1{o) ~(-J)--==II ~ 2+/2-3 ":'/1 #1/=-111/

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Download Quiz 01 - Solving Systems of Linear Equations through Row Operations and more Exercises Linear Algebra in PDF only on Docsity!

Math 205B (^) Quiz 01 page 1 01/15/

Name~'t *~ r./fip.

1. Considerthe augmented matrix A = [1~ -i 11~~].

1A. What system of linear equations is represented by A?

/h'X I - I.-/xz.~/OD

1.,., .,...><1.- ZL(

lB. Find the reduced row echelon form (rref) of A by hand, showing all the matrires produced along the way. Say what each elementary row operation was used to produce each new matrix from the previous one using the "rs +- r3 + 4r2" style notation we developed in class)

~ At< I!J&Yf..JJ. .fI!~ j """1 .a ,,*,:.1.""'" led ,b ,f iho< /Y''l/;J. lIdl'i // - :

[

I, ''

/

'OO

J

[

'-I -I 7-S'"

J

[

I -2.

J

r--

[

I -2.

I

3 I 2'( 3 f 2.'1 3 I l&j r.~r.-'r. 1. t), 0? 21 J

;:~rLlr

[

t -l..

I

17 r,~ t; ...t,1.

[

, D

I

r--Y 0 I 3J ~ D I 3 J

1C. What is the solution of the system in part 1A?

[

'I. =; t

x\ =3 '

Q dJ

z. 1 3 -3 11 11

2. Consider the system of equations whose augmented matrix is B = rrD 2 5 12 8 ~.

~~)j 1 13 13 11 ~

Use your calculator to find rref(B) and write the result in the space below. Circle the pivot columnsof B. Then use the
rref to find all the solutions of the system using the methods and notation we've developedin class. If there are no solutions,
explain why. Otherwise, give a specificsolution and show it satisfiesthe first equation the augmented matrix represents.

[

I 0 J 2.

/

O

J

7L kt ~ r~ Ii ~v~

rrt{( IS/ = ~ ~ -~ ~ f OxI r Dx. ~ 0. I ~ Clx, "" 1 ) ,A,d. ~ !!l s/cf1i,,~.

l' of / "",,11M

(su.^ L^ p'-vet.cJt g)^ ~~^ nil. M ~~ ,J. syd:1Ylr~'. "" .L.^ 11;^ ~^ UK

~ fk. fjsf£,(YI f(J IflCrfrt,Id~t...

[J

1

W

11

]

3. Now consider the system of equations whose augmented matrix is C = : 3 2 5 2 8.

: 13 1 11

Again, use your calculator to find rref( C) and write the result in the space below. Circle the pivot columns of C. Finally, use the rref to find all the solutions of the system using the methods and notation we've developed in class. H there are no solutions, explain why; otherwise, give a specific solution and show it satisfies the first equation the augmented matrix represents.

[

I J

( 0 3 0 2. )( -= 2 3- X

Jkrt) rrt{(C)::: D l -l. () -'" ~ Wt.fa ,,' - l.i

1

l ' () () 0 , .3. "z - I + X J

\ \ I )<.1 ~ f;.u.

FWJvMII$ ~l. c.) )<~ == -)

Skt. )( It, ~ IL. ~ oQ- mart" .rJ~{. 74/u Xj:: 0 t/W ~ !jl (2) 'fI OJ - j ) ~

1 ,/ ,p.

tJMsPtc,ihl.sJJW;.. l1k /11vIfckJr It in fk tI~VMl;o X I +-J X1.-1~ J + ~'1 = II : f 7 7 /' 2 i-3(f4) -1{o) ~(-J)--==II ~ 2+/2-3 ":'/1 # 1/=-111/