Chapter1:SystemsofLinearEquationsandMatrices
MultipleChoiceQuestions
1. Whichofthefollowingequationsislinear?
(A)2x2 1 +3x3 2 +4x4 3 =5
(B) √3x1 √2x2 + x3 =5
(C) √5x1 +5√x2 x3 =1
(D)22 x1 +cos(x2 )+4x3 =7
2. Whichsystemcorrespondstothefollowingaugmentedmatrix?
(A) x1 +11x2 = 3 9x1 +4x2 = 2
(B) x1 +11x2 +6x3 =3 9x1 +4x2 = 2
(C) x1 +11x2 +6x3 +3x4 =0 9x1 +4x2 2x4 =0
(D) x1 +9x2 =0 11x1 +4x2 =0 6x1 =0 3x1 2x2 =0
3. Whichofthefollowingstatementsbestdescribesthefollowingaugmentedmatrix?
(A) A isconsistentwithauniquesolution.
(B) A isconsistentwithinfinitelymanysolutions.
(C) A isinconsistent.
(D)noneoftheabove.
4. Whichofthefollowingmatricesisin reduced rowechelonform? (A)
⎢ ⎣ 10 11 0120 0131 ⎤ ⎥ ⎥
(B) ⎡ ⎢ ⎢ ⎣ 1025 01 75 00114
⎥ ⎥ ⎦ (C) 10011 3 00014
10 5 013 000
5. Ifthematrix A is4 × 2, B is3 × 4, C is2 × 4, D is4 × 3,and E is2 × 5,whichofthe followingexpressionsis not defined?
(A) AT D + CB T (B)(B + D T )A (C) CA + CB T (D) DBAE
6. Whatisthesecondrowoftheproduct AB ?
(A) 183325 (B) 644891 (C) 508999 (D) 488933
7. Whichofthefollowingisthedeterminantofthe2 × 2matrix A = ab cd ?
(A) ad bc (B) bc ad (C) 1 bc ad (D) 1 ad bc
8. Whichofthefollowingmatricesis not invertible?
(A) 36 24 (B) 77 23 (C) 90 44 (D) 93 65
9. Whichofthefollowingmatricesis not anelementarymatrix?
10 51 (B) 11 02 (C) ⎡
10. Forwhichelementarymatrix E willtheequation EA = B hold?
11. Whichmatrixwillbeusedastheinvertedcoefficientmatrixwhensolvingthefollowing system?
12. Whatvalueof b makesthefollowingsystemconsistent?
(A) b = 1(B) b =0(C) b =1(D) b =2
13. If A isa3 × 3diagonalmatrix,whichofthefollowingmatricesis not apossiblevalue of Ak forsomeinteger k ?
14. Thematrix
(A)uppertriangular. (B)lowertriangular. (C)both(A)and(B). (D)neither(A)nor(B).
15. If A isa4 × 5matrix,findthedomainandcodomainofthetransformation TA (x)= Ax.
(A)Notenoughinformation
(B)Domain: R4 ,Codomain: R5
(C)Domain: R5 ,Codomain: R5
(D)Domain: R5 ,Codomain: R4
16. Whichofthefollowingisamatrixtransformation?
(A) T (x,y,z )=(yx2 ,yz 2 )
(B) T (x,y,z,w )=(xy,yz,zw,wx)
(C) T (x,y,z )=(x +1,x +2,x + z,y + z )
(D) T (x,y )=(4x, 5x, x, 0)
17. Whichmatrixrepresentsreflectionaboutthe xy -plane?
18. Usematrixmultiplicationtofindtheimageofthevector 2, 1 whenitisrotated counterclockwiseabouttheoriginthroughanangle θ =45◦ .
19. Whichofthefollowingpairsofoperators T1 ,T2 : R2 → R2 commute?(Thatis,for whichpairisittruethat T1 ◦ T2 = T2 ◦ T1 ?)
(A) T1 isthereflectionaboutthe x-axis.
T2 isthereflectionaboutline y = x
(B) T1 istheorthogonalprojectionontothe x-axis.
T2 isthereflectionaboutline y = x.
(C) T1 isthecounterclockwiserotationabouttheoriginthroughanangleof π
T2 istheprojectionontothe y -axis.
(D) T1 isthereflectionaboutthe x-axis.
T2 isthecounterclockwiserotationabouttheoriginthroughanangleof π/2.
FreeResponseQuestions
1. Findtherelationshipbetween a and b suchthatthefollowingsystemhasinfinitelymany solutions.
x +2y = a 3x +6y = b
2. Solvethefollowingsystemanduseparametricequationstodescribethesolutionset.
3. Determinewhetherthefollowingsystemhasnosolution,exactlyonesolution,orinfinitely manysolutions.
4. Findthevalueof k thatmakesthesystem 15 36 10 k 9 inconsistent.
5. SolvethefollowingsystemusingGaussianelimination.
6. Solvethefollowingsystemfor x, y ,and z
7. Thecurve y = ax3 + bx2 + x + c passesthroughthepoints(0, 0), (1, 1), and( 1, 2). Findandsolveasystemoflinearequationstodeterminethevaluesof a,b, and c
8. Solvethefollowingsystemfor x and y x2 + y 2 =6 x2 y 2 =2
9. Given C = 1 1 20 ,find CC T
10. Expressthefollowingmatrixequationasasystemoflinearequations.
11. Findthe3 × 3matrix A =[aij ]whoseentriessatisfythecondition aij = i2 j .
12. Let A and B be n × n matrices.Provethattr(c A B )= c tr(A) tr(B ).
13. Whatistheinverseof 40 92 ?
14. Giventhepolynomial p(x)= x2 3x +1andthematrix A = 44 61 ,compute p(A).
15. Let A,B,C, and D be n × n invertiblematrices.Solvefor A giventhatthefollowing equationholds.
C 2 DA 1 CB 1 = BCB 1
16. Provethatforany m × n matrices A and B ,(A B )T = AT B T
17. Usetheinversionalgorithmtofindtheinverseofthefollowingmatrix.
18. Whichelementaryrowoperationwilltransformthefollowingmatrixintotheidentity matrix?
19. Findthe3 × 3elementarymatrixthatadds c timesrow3torow1.
20. Findtheelementarymatrix E thatsatisfies
21. Solvethefollowingsystembyinvertingthecoefficientmatrix.
22. Solvethefollowingmatrixequationfor X .
23. Giventhat A
24. Findanonzerosolutiontothefollowingequation.
25. Findthevaluesof a, b,and c thatmakethefollowingmatrixsymmetric.
26. Let A =
Findthediagonalentries c11 ,c22 , and c33 .
27. Lettheentriesofamatrix A =[aij ]bedefinedas aij =2i2 i + j + g (j ),where g isa functionof j .If A isasymmetricmatrix,whatis g (j )?
28. Provethatforanysquarematrix A,thematrix B =(A + AT )issymmetric.
29. Findthedomainandcodomainofthetransformationdefinedby
30. Findthestandardmatrixfortheoperator T : R2 → R2 definedby
31. Findthestandardmatrixforthetransformation T definedbytheformula
32. Findthestandardmatrix A forthelineartransformation T : R 2 → R2 forwhich
33. Provethatthecompositionoftworotationoperatorsabouttheoriginof R 2 isanother rotationabouttheorigin.
34. Provethatif TA : R3 → R3 and TA (x)= 0 foreveryvector x in R3 ,then A isthe3 × 3 zeromatrix.
35. Writeabalancedequationforthefollowingchemicalreaction.
36. Findthequadraticpolynomialwhosegraphpassesthroughthepoints 0, 3 , 1, 8 , and 1, 0
37. Usematrixinversiontofindtheproductionvector x thatmeetsthedemand d forthe consumptionmatrix C
FreeResponseAnswers 1. 3a = b
2. x1 = t +3, x2 = t +4, x3 = t
3. nosolution
4. k =2
5. x1 =5, x2 =11, x3 = 1
6. x =1, y = 1 2 , z = 1 3
7. System: c =0 a + b + c =0 a + b + c = 1
Solution: a = 1 2 ,b = 1 2 ,c =0
8. x = ±2, y = ±√2
9. CC T = 22 24
10. x +7y =0 4y +3z =0 6x 2z =0
11. A = ⎡ ⎢ ⎢ ⎣ 0 1 2 321 876 ⎤ ⎥ ⎥ ⎦ 13. 1 4 0 9 8 1 2 14. 298 1223
15. A = B 1 C 2 D
18. Add9timesrow2torow4
19. ⎡ ⎢ ⎢ ⎣ 10 c 010 001 ⎤ ⎥ ⎥ ⎦
20. E = ⎡ ⎢ ⎢ ⎣ 100 010 201 ⎤ ⎥ ⎥ ⎦
21. x = 9, y =32
22. ⎡ ⎢ ⎢ ⎣ 33 43 6723
23.x = ⎡ ⎢ ⎢ ⎣ 3 9 4 ⎤ ⎥ ⎥ ⎦
24. Any x = x1 x2 suchthat2x1 =3x2 .Possiblesolution: x = 3 2
25. a =4,b =0,c =4
26. c11 = 10, c22 =0,and c33 =4
27. g (j )=2j 2 2j
29. Domain: R4 ,Codomain: R2
30. 31 04
⎡
32. 813 20
35. C3 H8 +5O2 → 4H2 O+3CO2
36. 3 4x + x2
37.x ≈ ⎡ ⎢ ⎢ ⎣ 91 85 125.50 135 10 ⎤ ⎥ ⎥ ⎦