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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.
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