Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath 45—Linear AlgebraQuiz #4—Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: October 3, 2001 Version 1.002Directions Place your solution in the space provided! No calculators allowed!Exercise 1. Let A be an n × n matrix. Prove or disprove: A − ATis symmetric.Exercise 2. GivenA =12 320−43 −42,factor A as A = LDLT.Solutions to Exercises 3Solutions to ExercisesExercise 1. The statement is false. For a counterexample, consider the matrixA =1234.ThenA − AT=1234−1324=0 −110.Exercise 1Exercise 2. Use the pivot in row 1 column 1 to find the multipliers for elimination.l21= a21/a11=2/1=2l31= a31/a11=3/1=3Store these in a lower triangular matrix L.L =100210301Subtract 2 times row 1 from row 2 of matrix A. Subtract 3 times row 1 from row 3 of matrix A.E31E21A =12 30 −4 −100 −10 −7Use the pivot in row 2 column 2 of matrix E31E21A to find the next multiplier.l32= a∗32/a∗22= −10/ − 4=5/2Store this multiplier in the matrix L.L =10021035/21Subtract 5/2 times row 2 from row 3 of matrix E31E21A.E32E31E21A =12 30 −4 −1000 18= UTherefore,A = LU =10021035/2112 30 −4 −1000 18.To complete the factorization, factor a diagonal matrix D from U.A = LDLT=10021035/2110 00 −40001812 3015/200 1Exercise
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