Exercises_AppliedLinearAlgebra3.pdf - Exercises on Applied...

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Exercises on Applied Linear AlgebraDuong Thanh PHAMNovember 18, 20191Systems of Linear Equations and Matrices1.1Systems of linear equations and Gaussian eliminationExercise 1.Solve the following systems1.8>>><>>>:2x1-x2-x3= 43x1+ 4x2-2x3= 113x1-2x2+ 4x3= 112.8>>><>>>:x1+x2+ 2x3=-12x1-x2+ 2x3=-44x1+x2+ 4x3=-23.8>>>>>><>>>>>>:x1+ 2x2+ 3x3-2x4= 62x1-x2-2x3-3x4= 43x1+ 2x2-x3+ 2x4= 42x1-3x2+ 2x3+x4=-8Exercise 2.Solve the following systems1.8>>><>>>:mx1+x2+x3= 1x1+mx2+x3=mx1+x2+mx3=m22.8>>><>>>:(m+ 3)x1+x2+ 2x3=mmx1+ (m-1)x2+x3= 2m3(m+ 1)x1+mx2+ (m+ 3)x3= 3Exercise 3.Solve the following systems by using Gaussian Elimination1.8>>>>>><>>>>>>:x+y+z+t+u= 73x+ 2y+z+t-3u=-2y+ 2z+ 2t+ 6u= 235x+ 4y+ 3z+ 3t-u= 122.8>>>>>><>>>>>>:2x+y-z-t+u= 1x-y+z+t-2u= 03x+ 3y-3z-3t+ 4u= 24x+ 5y-5z-5t+ 7u= 33.8>>>>>><>>>>>>:2x-2y+z-t+u= 1x+ 2y-z+t-2u= 14x-10y+ 5z-5t+ 7u= 12x-14y+ 7z-7t+ 11u=-14.8>>>>>><>>>>>>:3x+y-2z+t-u= 12x-y+ 7z-3t+ 5u= 2x+ 3y-2z+ 5t-7u= 33x-2y+ 7z-5t+ 8u= 3Exercise 4.Solve the corresponding homogeneous systems of the systems in Exercise 3.1
1.2MatricesExercise 5.Find transposes of the following matrices:1.I2,I3,In2. 032,O32,Omn3.A=2666641-11800-2137006-23000173777754.B=2643-1081497512015-10-64801103755.C=266664-50000-20000600007377775Exercise 6.Find the unknown entries of the following matrices:1.A=26408-3x-12yz11375is a symmetric matrix,2.B=2666643x2y-19011zt-3620uv8377775is a skew-symmetric matrix,Exercise 7.FindA+BandA-Bof the following matrices:1.A=h10-39108iandB=h04-3-60-20i2.A=266666645-2239037777775andB=266666641040510377777753.A="12-1312#andB="12331-1#Exercise 8.Find the unknown entries of the following matrices1. 326408-3x-12yz33375=264024-90-3617-3113752.13266664x32013-19011zt-3620uv8377775=2666643w20/3y-1/33011/3107mn20/3610k377775Exercise 9.LetA="12-1312#,B=26413213-1375andC="2-110#. ComputeAB, BA, AC, CA, BCandCBwhenever they are well-defined.2
Exercise 10.Compute the products1.2644010104013752643453752.2641000100013752645-233753."2013# "11#Exercise 11.Compute1.h1-27i2641-273752.2641-27375h1-27iExercise 12.Find 22 matrices such that1.A2=-I,Ahaving only real entries2.B2= 0, althoughB6= 023.CD=-DC, whereCD6= 024.EF= 0, in which no entries ofEandFare zero.Exercise 13.Which of the following matrices equal (A+B)2?A2+ 2AB+B2,A(A+B) +B(A+B), (A+B)(B+A),A2+AB+BA+B2.Exercise 14.FindA2, A3;B2, B3andC2, C3. What areAk, BkandCk?A="12121212#,B="100-1#andC=AB.Exercise 15.Find1."2113#32."32-4-2#53."1101#n4."λ10λ#n5.

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