Linear Algebra

Section IV. Matrix Operations 2314.22 Do the calculations for the proof of Corollary 4.12.4.23 Show that this matrix ( )1 0 1H =0 1 0has infinitely many right inverses. Show also that it has no left inverse.4.24 In Example 4.1, how many left inverses has η?4.25 If a matrix has infinitely many right-inverses, can it have infinitely manyleft-inverses? Must it have?̌ 4.26 Assume that H is invertible and that HG is the zero matrix. Show that G isa zero matrix.4.27 Prove that if H is invertible then the inverse commutes with a matrix GH −1 =H −1 G if and only if H itself commutes with that matrix GH = HG.̌ 4.28 Show that if T is square and if T 4 is the zero matrix then (I − T ) −1 =I + T + T 2 + T 3 . Generalize.̌ 4.29 Let D be diagonal. Describe D 2 , D 3 , . . . , etc. Describe D −1 , D −2 , . . . , etc.Define D 0 appropriately.4.30 Prove that any matrix row-equivalent to an invertible matrix is also invertible.4.31 The first question below appeared as Exercise 28.(a) Show that the rank of the product of two matrices is less than or equal tothe minimum of the rank of each.(b) Show that if T and S are square then T S = I if and only if ST = I.4.32 Show that the inverse of a permutation matrix is its transpose.4.33 The first two parts of this question appeared as Exercise 25.(a) Show that (GH) trans = H trans G trans .(b) A square matrix is symmetric if each i, j entry equals the j, i entry (that is, ifthe matrix equals its transpose). Show that the matrices HH trans and H trans Hare symmetric.(c) Show that the inverse of the transpose is the transpose of the inverse.(d) Show that the inverse of a symmetric matrix is symmetric.̌ 4.34 The items starting this question appeared as Exercise 30.(a) Prove that the composition of the projections π x , π y : R 3 → R 3 is the zeromap despite that neither is the zero map.(b) Prove that the composition of the derivatives d 2 /dx 2 , d 3 /dx 3 : P 4 → P 4 isthe zero map despite that neither map is the zero map.(c) Give matrix equations representing each of the prior two items.When two things multiply to give zero despite that neither is zero, each is said tobe a zero divisor. Prove that no zero divisor is invertible.4.35 In real number algebra, there are exactly two numbers, 1 and −1, that aretheir own multiplicative inverse. Does H 2 = I have exactly two solutions for 2×2matrices?4.36 Is the relation ‘is a two-sided inverse of’ transitive? Reflexive? Symmetric?4.37 Prove: if the sum of the elements of a square matrix is k, then the sum of theelements in each row of the inverse matrix is 1/k. [Am. Math. Mon., Nov. 1951]