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Math 225 Chapter 3 Exercise Sets

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California State University, East Bay**We aren't endorsed by this school
Course
MATH 225
Subject
Mathematics
Date
Dec 16, 2024
Pages
7
Uploaded by ElderSardineMaster814
Math 225Brandon FitzgeraldSection 3.1 Exercise Set1-12) Simplify the following expressions with matrices by performing the matrix operations.1) [8−174]+[−32−511]2) [103−3−6]−[7−326]3) [9241−9−2−65−3]+[2−4−7143−25−1]4) [−37−761083−87]−[−1341521−2−5]5) [710−3−4−78]+[1−48−1−49]6) [61687−2−7]−[−345−23−1]7) [61−511]+[−21273]8) [11753]−[7532]9) 4∙[1352]+3∙[7−34−1]10) 5∙[−244−2]−2∙[1331]11) −3∙[200010005]+2∙[−100020007]12) 5∙[400020001]−3∙[100040009]13-16) Identify if the following matrices can be multiplied based on their dimensions and if they can be multiplied, find what the dimensions of the product would be.13) [4218−65]∙[2−4−7143]14) [2−138]∙[−1−3823−1]15) [−73173−523]∙[125−8]16) [5−713]∙[5−73]17-22) Simplify the following expressions with matrices by performing the matrix operations.17) [3−174]∙[−32−53]18) [1331]∙[−244−2]19) [4215−35]∙[2−4−2123]20) [2−4−2123]∙[4215−35]21) [1234]∙[1234]22) [1234]∙[1234]23) −4∙[3−123]∙[12]+[47]24) 2∙[23−14]∙[7−2]−[31]25) 3∙[204315167]∙[2−31]+[−478]26) 4∙[−2122−31205]∙[−103]−[31−2]
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Math 225Brandon FitzgeraldSection 3.2 Exercise Set1-4) For the following vectors,(a) Graph the vector(b) Find the Euclidean norm of the vector.1)[27]2)[-58]3)[-6-8]4)[1-6]5-12) For the following vectors, find the(a) Euclidean Norm(b) Taxicab norm(c) Max norm5)[236]6)[-14-8]7)[-523]8)[7-31]9)[1-534]10)[- 22- 47]11)[6-125]12)[-82- 41]13-22) Perform the following operations on vectors.13)[5-3]+[26]14)[92]−[2- 4]15)−3∙[61]16)4∙[- 17]17)[41-3]−[-56- 2]18)[7- 26]+[- 385]19)4∙[94- 3]20)-2∙[1-78]21)2∙[1593]−[2631]22)-3∙[71- 24]+[-3982]23) Show that for a 3-dimensional vector x=[x1x2x3]and a constant c, the taxicab norm satisfies the property that scalar multiplication scales the norm of the vector, i.e., show that||c∙ x||1=|c|∙||x||1.24) Show that for a 3-dimensional vector x=[x1x2x3]and a constant c, the Euclidean norm satisfies the property that scalar multiplication scales the norm of the vector, i.e., show that||c∙ x||2=|c|∙||x||2.25-28) Given the magnitude of a vector, find the magnitude of a constant multiple of that vector25) Given that ||x||=5, find ||7x||.26) Given that ||x||=6 , find ||-2x|| .27) Given that ||x||=6 , find ||3x|| .28) Given that ||x||=7 , find ||- 4x|| .
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Math 225Brandon FitzgeraldSection 3.3 Exercise Set1-2) Find the linear transformation of the following vectors using the given matrix.1) For T(x)=A xand A=[4152], find:(a)T([2-3])(b)T([-13])2) For T(x)=A xand A=[216-1321-52], find:(a)T([121])(b)T([-312])3-5) Graph the transformation of the unit square for the following matrices.3) A=[3122]4) A=[0- 220]5) A=[-1032]6-8) Apply the following shear transformations to the stated vectors.6) Apply a scaling transformation to the vector [3-5]to scale it horizontally by a factor of 4 andvertically by a factor of 3.7) Apply a shear transformation in the ydirection with an angle of θ=π6to the vector [14].8) Apply a rotation transformation with an angle of θ=5π6to the vector [-37].9-11) Find a single matrix that will apply the following collection of transformations in the order given.9) Reflects a vector about the y-axis, rotates it by θ=2π3, and then scales it vertically by a factor of 3.10) Shears a vector in the ydirection with an angle of θ=5π6, reflects it about the y-axis, and thenscales it horizontally by a factor of 4.11) Rotates a vector by θ=3π4, reflects it about the x-axis, and then shears it in the xdirection with anangle of θ=π3.12) Show that reflection about the x-axis then reflection about the y-axis is equivalent to rotation by θ=π.13) In general, matrix multiplication of square matrices is not commutative (A∙B≠B∙A). However, for specific types of matrices, it may be true that A∙B=B∙A. Show that multiplication of scaling matrices and reflection about the x-axisiscommutative.
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Math 225Brandon FitzgeraldSection 3.4 Exercise Set1-6) Find the determinants of the following matrices. 1)[234-8]2)[8-237]3)[4-3-86]4)[3846]5)[-516-3]6)[4-82-1]7-12) Find the determinants of the following matrices by using cofactor expansion. 7)[4-3012- 43-25]8)[2082-14-15- 3]9)[219- 40683- 1]10)[- 24-14- 14-3- 5- 2]11)[3- 2811-322-6]12)[-5642145-5- 4]13-18) Find the determinants of the following matrices by using the Rule of Sarrus.13)[4-3012- 43-25]14)[2082-14-15- 3]15)[219- 40683- 1]16)[- 24-14- 14-3- 5- 2]17)[3- 2811-322-6]18)[-5642145-5- 4]
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Math 225Brandon FitzgeraldSection 3.1 Exercise Set Answers1) [51215]2) [36−5−12]3) [11−2−32−51−810−4]4) [−24−115562−612]5) [865−5−1117]6) [91239−5−6]7) [413214]8) [4221]9) [253325]10) [−121414−12]11) [−80001000−1]12) [17000−2000−22]13) Can be multiplied to obtain a 3⨯3matrix.14) Can't be multiplied.15) Can't be multiplied16) Can be multiplied to obtain a 4⨯3 matrix.17) [−43−4126]18) [10−2−210]19) [10−12−27613−12221]20) [10−26−327]21) [1234246836912481216]22) [30]23) [025]24) [13−31]25) [2031-19]26) [29354]Section 3.2 Exercise Set Answers1-4)1) (a)(b) √532) (a)(b) √89 3) (a)(b) 104) (a)(b) √37
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Math 225Brandon Fitzgerald5) (a) 7 (b) 11(c) 66) (a) 9 (b) 13(c) 87) (a) √38 (b) 10(c) 58) (a) √59 (b) 11(c) 79) (a) √51 (b) 13(c) 510) (a) √73 (b) 15(c) 711) (a) √56 (b) 14(c) 612) (a) √85(b) 15 (c) 813) [73]14) [76]15) [-18-3]16) [- 428]17) [9-5-1]18) [4611]19) [3616-12]20) [- 214-16]21) [04155]22) [- 24614-10]23-24) Sorry about the poor formatting, the outside norm bars (double vertical bars) should be taller.23) Hint: You should be showing that ∥c ∙[x1x2x3]∥1=∙∙∙=|c|∙∥[x1x2x3]∥1.23) Hint: You should be showing that ∥c ∙[x1x2x3]∥2=∙∙∙=|c|∙∥[x1x2x3]∥2.25) 3526) 1227) 1828) 28Section 3.3 Exercise Set Answers1) (a)[54](b)[-11]2) (a)[107-7](b)[710- 4]3) 4)5)6) [12-15]7) [14+√33]≈[14.58]8) [-72+3√32-32−7√32]≈[-0.90-7.56]
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Math 225Brandon Fitzgerald9) [12-√32-3√32-32]≈[0.5- 0.87- 2.60-1.5]10) [- 40-√331]≈[- 40- 0.581]11) [-√62−√22√62−√22-√22√22]≈[-1.930.52-0.710.71]12) The answer is technically the entire work to show that it is true. You should be showing that whenyou multiply a reflection about the x-axis matrix and a reflection about the y-axis matrix, you get thesame thing as a rotation by θ=πmatrix.13) The answer is technically the entire work to show that it is true. You should be showing that for ageneral scaling matrix, A, and a reflection about the x-axis matrix, B, A∙B=B∙A.Section 3.4 Exercise Set Answers1)−282)623)04)−145)96)127)598)389)−10010)−3711)012)2813)5914)3815)−10016)−3717)018)28
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