Solutions to Section 2 1,True False Review, a TRUE A diagonal matrix has no entries below the main diagonal so it is upper triangular Likewise. it has no entries above the main diagonal so it is also lower triangular. b FALSE An m n matrix has m row vectors and n column vectors. c TRUE This is a square matrix and all entries o the main diagonal are zero so it is a diagonal matrix. the entries on the diagonal also happen to be zero but this is not required. d FALSE The main diagonal entries of a skew symmetric matrix must be zero In this case a11 4 0. so this matrix is not skew symmetric, e FALSE The form presented uses the same number along the entire main diagonal but a symmetric. matrix need not have identical entries on the main diagonal. f TRUE Since A is symmetric A AT Thus AT T A AT so AT is symmetric. g FALSE The trace of a matrix is the sum of the entries along the main diagonal. h TRUE If A is skew symmetric then AT A But A and AT contain the same entries along the. main diagonal so for AT A both A and A must have the same main diagonal This is only possible. if all entries along the main diagonal are 0, i TRUE If A is both symmetric and skew symmetric then A AT A and A A is only possible. if all entries of A are zero, j TRUE Both matrix functions are de ned for values of t such that t 0. k FALSE The 3 2 entry contains a function that is not de ned for values of t with t 3 So for. example this matrix functions is not de ned for t 2. l TRUE Each numerical entry of the matrix function is a constant function which has domain R. m FALSE For instance the matrix function A t t and B t t2 satisfy A 0 B 0 but A and. B are not the same matrix function,1 a a31 0 a24 1 a14 2 a32 2 a21 7 a34 4. 1 b 1 4 and 3 2,2 a b12 1 b33 4 b41 0 b43 8 b51 1 and b52 9. 2 b 1 2 1 3 2 1 3 2 and 5 1,1 3 2 2 matrix,2 3 matrix. 1 4 1 matrix,4 3 matrix,7 1 0 3 3 3 matrix,8 4 4 matrix. 9 4 4 matrix,10 tr A 1 3 4,11 tr A 1 2 3 0,12 tr A 2 2 5 1. 13 Column vectors,Row vectors 1 1 3 5,14 Column vectors 1 2 5. Row vectors 1 3 4 1 2 5 2 6 7,15 Column vectors,Row vectors 2 10 6 5 1 3. 16 A 3 4 Column vectors 3 4,2 0 4 1 1 2 0 4 1 1,17 A column vectors. 9 4 4 0 8 9 4 4 0 8,18 B 3 0 row vectors 2 4 6 6 3 0 1 0 2 1. 19 B 1 7 0 2 Row vectors 2 5 0 1 1 7 0 2 4 6 0 3, 20 A a1 a2 ap has p columns and each column q vector has q rows so the resulting matrix has. dimensions q p,21 One example 0 3 0,22 One example 0. 23 One example,24 One example 0 2 0, 25 The only possibility here is the zero matrix 0 0 0. 27 One example 0 0,28 One example 3 t,29 One example. 30 One example t2 1 1 1 1 1, 31 One example Let A and B be 1 1 matrix functions given by. A t t and B t t2, 32 Let A be a symmetric upper triangular matrix Then all elements below the main diagonal are zeros. Consequently since A is symmetric all elements above the main diagonal must also be zero Hence the. only nonzero entries can occur along the main diagonal That is A is a diagonal matrix. 33 Since A is skew symmetric we know that aij aji for all i j But since A is symmetric we know. that aij aji for all i j Thus for all i j we must have aji aji That is aji 0 for all i j That. is every element of A is zero,Solutions to Section 2 2. True False Review, a FALSE The correct statement is AB C A BC the associative law A counterexample to the. particular statement given in this review item can be found in Problem 5. b TRUE Multiplying from left to right we note that AB is an m p matrix and right multiplying AB. by the p q matrix C we see that ABC is an m q matrix. c TRUE We have A B T AT B T A B so A B is symmetric. 0 1 0 0 0 3, d FALSE For example let A 1 0 0 B 0 0 0 Then A and B are skew. 0 0 0 3 0 0,symmetric but AB 0 0 3 is not symmetric. e FALSE The correct equation is A B 2 A2 AB BA B 2 The statement is false since AB BA. 1 0 0 1 2 1 1, does not necessarily equal 2AB For instance if A and B. 0 0 0 0 then A B 0 0,and A2 2AB B 2 A B 2, f FALSE For example let A and B Then AB 0 even though A 0 and. g FALSE For example let A and let B Then A is not upper triangular. despite the fact that AB is the zero matrix hence automatically upper triangular. h FALSE For instance the matrix A is neither the zero matrix nor the identity matrix. and yet A2 A, i TRUE The derivative of each entry of the matrix is zero since in each entry we take the derivative. of a constant thus obtaining zero for each entry of the derivative of the matrix. j FALSE The correct statement is given in Problem 45 The problem with the statement as given is. that the second term should be dA dA,dt B not B dt. k FALSE For instance the matrix function A satis es A but A does not have. l TRUE This follows by exactly the same proof as given in the text for matrices of numbers see part. 3 of Theorem 2 2 23,1 c iC 1 3i 1 4i,1 e A 3C T,1 3i 15 3i 16 3i. 5 3i 12 3i 15 3i, Differential Equations and Linear Algebra 4th Edition Goode. SOLUTIONS MANUAL,Full download at, https testbankreal com download differential equations linear algebra 4th. edition goode solutions manual, differential equations and linear algebra goode 4th edition pdf. differential equations and linear algebra 4th edition by goode and annin pdf. differential equations and linear algebra 4th edition solutions. differential equations and linear algebra 4th edition pdf. differential equations and linear algebra 4th edition edwards pdf. differential equations and linear algebra goode pdf. differential equations and linear algebra 4th edition slader. differential equations and linear algebra 4th pdf,.

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