Linear Algebra, 2020a

Section IV. Matrix Operations 261
4.25 Assume that g: V → W is linear. One of these is true, the other is false. Which
is which?
(a) If f: W → V is a left inverse of g then f must be linear.
(b) If f: W → V is a right inverse of g then f must be linear.
̌ 4.26 Assume that H is invertible and that HG is the zero matrix. Show that G is a
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 30.
(a) Show that the rank of the product of two matrices is less than or equal to the
minimum of the rank of each.
(b) Show that if T and S are square then TS = I if and only if ST = I.
4.32 Show that the inverse of a permutation matrix is its transpose.
4.33 (a) Show that (GH) T = H T G T .
(b) A square matrix is symmetric if each i, j entry equals the j, i entry (that is,
if the matrix equals its transpose). Show that the matrices HH T and H T H are
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 (a) Prove that the composition of the projections π x ,π y : R 3 → R 3 is the zero
map 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 is the
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
to be a zero divisor. Prove that no zero divisor is invertible.
4.35 In the algebra of real numbers, quadratic equations have at most two solutions.
Matrix algebra is different. Show that the 2×2 matrix equation T 2 = I has more
than two solutions.
4.36 Is the relation ‘is a two-sided inverse of’ transitive? Reflexive? Symmetric?
4.37 [Am. Math. Mon., Nov. 1951] Prove: if the sum of the elements of each row
of a square matrix is k, then the sum of the elements in each row of the inverse
matrix is 1/k.