Appendix A — Answers to Odd-Numbered Exercises

Answers, with the deciding step — not full solutions. If your answer differs, re-derive before consulting; if it matches, make sure it would have matched for the reason given.

§1.1 Kinds of Numbers

1. (a) 6 \in \mathbb{N} — a count. (b) +7 \in \mathbb{N} as a value, though oxidation states as a type live in \mathbb{Z} (they can be negative — the smallest system containing this one value and the smallest system appropriate to the quantity differ). (c) \tfrac{3}{8} \in \mathbb{Q} — a ratio of counts. (d) \pi \in \mathbb{R} — irrational.

3. 100x - x = 27.\overline{27} - 0.\overline{27} = 27, so 99x = 27 and x = \tfrac{27}{99} = \tfrac{3}{11}.

5. If rs were rational, then s = (rs) \cdot \tfrac1r would be a product of rationals, hence rational — contradicting the hypothesis. So rs is irrational.

7. Mimic the parity argument mod 3: squares leave remainder 0 or 1 on division by 3, so p^2 = 3q^2 forces 3 \mid p, then 3 \mid q — contradiction with lowest terms. For \sqrt4: the step “p^2 divisible by 4 \Rightarrow p divisible by 4” is false (p = 2), because 4 is not prime — and indeed \sqrt4 = 2 is rational.

9. The false step is “no gaps between rationals \Rightarrow every point of the line is rational.” Density guarantees rationals arbitrarily close to any point; it does not guarantee the point itself is rational. \sqrt2 is approximated arbitrarily well by rationals and is provably not one of them — closeness is not membership.

§1.2 Sets and Collections

1. R \subseteq H: yes — every member of R is a halogen. \mathrm{Br} \in R: yes. \{\mathrm{Br}\} \in R: no — R’s members are elements, not sets (\{\mathrm{Br}\} \subseteq R would be the true statement).

3. |A \cup B| = 87 + 64 - 41 = 110 pass at least one; 200 - 110 = 90 fail both. Categories: 46 + 23 + 41 + 90 = 200 ✓.

5. 6 \times 4 \times 3 = 72 experiments. The number 13 counts the settings available on the three dials considered separately — e.g., a one-factor-at-a-time sweep from a fixed baseline — not the full combination grid.

7. \varnothing; \{2p_x\}, \{2p_y\}, \{2p_z\}; the three pairs; the full set — 8 = 2^3 subsets. In general each element is independently in or out: 2 \times 2 \times \cdots \times 2 = 2^n by the multiplication principle.

9. Audit the seven membership patterns. An element in A only: counted once (by |A|). In A \cap B only: 1 + 1 - 1 = 1. In all three: 3 - 3 + 1 = 1. Every pattern nets to exactly one count, so the right side equals |A \cup B \cup C|.

§1.3 Permutations and Combinations

1. (a) 20^4 = 160{,}000 (order matters, repetition allowed). (b) 5! = 120 (order matters, no repetition). (c) \binom{12}{4} = 495 (unordered panel).

3. \dfrac{9!}{4!} = \dfrac{362{,}880}{24} = 15{,}120 — the four E’s are interchangeable, so each distinct word was counted 4! times.

5. \binom{10}{2} = 45 microstates; the term degeneracies 21 + 5 + 9 + 9 + 1 = 45 ✓.

7. Algebraically, \binom{n}{n-k} = \frac{n!}{(n-k)!\,k!} = \binom{n}{k}. Bijectively: choosing which k solvents join the panel is choosing which n-k stay on the shelf — the same decision described from the other side.

9. Hydrogen pairs from \{\mathrm{H}, \mathrm{D}\} with equivalent positions: HH, HD, DD — 3 (not 4: HD = DH). Times 3 oxygen isotopes: 3 \times 3 = 9 isotopologues.

§1.4 Complex Numbers

1. (4 - 2i); (3+2i)(1-4i) = 3 - 12i + 2i - 8i^2 = 11 - 10i; i^3 = -i; i^4 = 1; i^{2026} = i^{4 \cdot 506 + 2} = i^2 = -1.

3. |3+4i| = 5, |1+i| = \sqrt2, |i| = 1; z\bar z = (3+4i)(3-4i) = 9 + 16 = 25. Jointly: z\bar z = |z|^2.

5. \varphi = 60°: 2 + 2(\tfrac12) = 3 — constructive side, partway toward full bonding reinforcement (which is 4, at \varphi = 0). \varphi = 120°: 2 + 2(-\tfrac12) = 1 — partial cancellation. \varphi = 180°: 0 — total annihilation, the antibonding node.

7. (ac-bd)^2 + (ad+bc)^2 = a^2c^2 - 2abcd + b^2d^2 + a^2d^2 + 2abcd + b^2c^2 = (a^2+b^2)(c^2+d^2) — the cross terms cancel, exactly as in §3.4’s Lagrange identity.

9. |z|^2 = |i| = 1 so |z| = 1; the angle must satisfy 2\theta = 90° + 360°k, giving \theta = 45° or 225°: z = \pm\tfrac{\sqrt2}{2}(1 + i). A degree-2 polynomial is entitled to exactly 2 roots in \mathbb{C} — both found.

§1.5 Working in the Complex Plane

1. 9 + 4 = 13; 25 + 1 = 26; 1 + 1 = 2. The identity: (a+bi)(a-bi) = a^2 + b^2 = |z|^2 — always real, never negative.

3. 1+i: (\sqrt2, 45°). -2: (2, 180°). 3i: (3, 90°). 1 - \sqrt3\,i: (2, 300°) (equivalently -60°).

5. (1+i)^{12} = (\sqrt2)^{12} \operatorname{cis}(12 \times 45°) = 64 \operatorname{cis} 540° = 64 \operatorname{cis} 180° = -64. Cross-check: (1+i)^4 = -4, and (-4)^3 = -64 ✓.

7. Both identities fall out of expanding z = a+bi, w = c+di on each side (the product case reuses the (ac-bd, ad+bc) rule with b, d negated). Then |zw|^2 = zw\,\overline{zw} = z\bar z\, w\bar w = |z|^2 |w|^2; take roots.

9. z = \dfrac{-1 \pm \sqrt{1 - 4}}{2} = -\tfrac12 \pm \tfrac{\sqrt3}{2} i. Both have modulus \sqrt{\tfrac14 + \tfrac34} = 1 and angles 120°, 240° — the two nontrivial cube roots of unity: cubing triples the angles to 360° and 720°, both \equiv 0°, so z^3 = 1. That is exactly right, since z^2 + z + 1 = \frac{z^3 - 1}{z - 1}: its roots are the cube roots of 1 other than 1 itself.

§2.1 Direction in Space

1. \mathbf{a}+\mathbf{b} = (-1, 4, 5); \mathbf{a}-\mathbf{b} = (5, -6, 3); 3\mathbf{a}-2\mathbf{b} = (12, -13, 10).

3. Vectors: bond dipole, force, velocity. Scalars: bond energy, pH, W — each is fully given by one number with no direction.

5. 2(1.52)\cos(54.75°) = 1.75 D — smaller than water’s 1.86 D: opening the angle moves the bond dipoles toward cancellation.

7. With \mu_b(1, 0), \mu_b(-\tfrac12, \tfrac{\sqrt3}{2}), \mu_b(-\tfrac12, -\tfrac{\sqrt3}{2}): the x-components sum to \mu_b(1 - \tfrac12 - \tfrac12) = 0 and the y-components cancel pairwise. Net dipole exactly \mathbf{0} — why BF₃, with three strongly polar bonds, is a nonpolar molecule.

9. |\mathbf{a}+\mathbf{b}|^2 = |\mathbf{a}|^2 + 2\,\mathbf{a}\cdot\mathbf{b} + |\mathbf{b}|^2 \le |\mathbf{a}|^2 + 2|\mathbf{a}||\mathbf{b}| + |\mathbf{b}|^2 = (|\mathbf{a}|+|\mathbf{b}|)^2 by Cauchy–Schwarz. Equality iff \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|, i.e. parallel and same sense: bond dipoles reach the sum of magnitudes only when collinear and aligned.

§2.2 Magnitude and Components

1. 5, 3, 7, 7. The last two must coincide before any arithmetic: squaring kills the signs, and reversing an arrow cannot change its length.

3. |\mathbf{v}| = 7; \hat{\mathbf{v}} = (\tfrac67, -\tfrac27, \tfrac37), whose squares sum to \tfrac{36+4+9}{49} = 1 ✓; \mathbf{v} = 7\,(\tfrac67, -\tfrac27, \tfrac37).

5. c\mathbf{v} = (-3, -6, -6), |c\mathbf{v}| = 9; and |c||\mathbf{v}| = 3 \times 3 = 9 ✓.

7. From |\hat{\mathbf{v}}| = 1: \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1. Three equal angles of 60°: 3 \times \tfrac14 = \tfrac34 \neq 1 — impossible. 55°: 3\cos^2 55° = 0.987 \neq 1 — also impossible. The unique equal-angle solution is \cos\alpha = 1/\sqrt3, \alpha = 54.74° — the “magic angle” of solid-state NMR.

9. Expanding with |\mathbf{v}|^2 = \mathbf{v}\cdot\mathbf{v} and cancelling \mathbf{r}\cdot\mathbf{r}: 2\,\mathbf{r} \cdot (\mathbf{b} - \mathbf{a}) = |\mathbf{b}|^2 - |\mathbf{a}|^2 — linear in \mathbf{r}: a plane, perpendicular to \mathbf{b} - \mathbf{a}, through the midpoint of the two atoms.

§2.3 The Dot Product

1. \mathbf{a}\cdot\mathbf{b} = 2 - 4 - 6 = -8; \mathbf{a}\cdot\mathbf{a} = 14. Negative dot product: obtuse.

3. (3,-2,1)\cdot(2,3,0) = 6 - 6 + 0 = 0: orthogonal. (1,1,1)\cdot(1,1,-1) = 1: not. No magnitudes needed — orthogonality is defined by a vanishing dot product, and the sign/zero of \mathbf{a}\cdot\mathbf{b} alone classifies the angle.

5. \mathbf{b}_1 \cdot \mathbf{b}_2 = -0.4418 + 0 + 0.1444 = -0.2974; |\mathbf{b}_1| = |\mathbf{b}_2| = 1.014; \cos\theta = -0.2893, \theta = 106.8° — matching experiment.

7. \operatorname{comp} = 0.9 D; |\boldsymbol{\mu}| = \sqrt{0.81 + 1.44} = 1.5 D; \cos\theta = 0.6, \theta = 53.1°.

9. |\hat{\mathbf{u}}-\hat{\mathbf{v}}|^2 = \hat{\mathbf{u}}\cdot\hat{\mathbf{u}} - 2\hat{\mathbf{u}}\cdot\hat{\mathbf{v}} + \hat{\mathbf{v}}\cdot\hat{\mathbf{v}} = 2(1 - \cos\theta). At \cos\theta = -\tfrac13: |\hat{\mathbf{u}}-\hat{\mathbf{v}}| = \sqrt{8/3} = 1.633.

§2.4 Basis and Coordinates

1. (a) Not a basis: (2,4) = 2(1,2) — the pair spans only a line; (1, 0) is unreachable, and (2,4) itself has two descriptions (1 \cdot second vector, or 2 \cdot first). (b) Basis. (c) Basis (orthonormal).

3. Orthonormality checks pass; c_1 = \tfrac{5+1}{\sqrt2} = 3\sqrt2 \approx 4.243, c_2 = \tfrac{5-1}{\sqrt2} = 2\sqrt2 \approx 2.828; c_1^2 + c_2^2 = 18 + 8 = 26 = 5^2 + 1^2 ✓.

5. Cubic cell vectors are mutually perpendicular (cross terms vanish) and of equal known length a (so dividing by |\mathbf{a}|^2 normalizes): u = \mathbf{v}\cdot\mathbf{a}/a^2 exactly. Orthogonality and known normalization are the two working properties; triclinic cells lack the first.

7. Dotting the equality with \hat{\mathbf{e}}_i kills every term but the i-th on each side, leaving c_i = c_i'. Orthonormality is what annihilates the cross terms; without it each dot product yields an equation mixing all the coefficients, and equality of individual coefficients can no longer be read off.

9. \mathbf{v}\cdot\mathbf{v} = \sum_i \sum_j c_i c_j\, \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j = \sum_i c_i^2 since only i = j survives. Any orthonormal frame therefore computes the same |\mathbf{v}|^2 from its own coordinates — Example 2.4.5’s check in general.

§3.1 Grids of Numbers

1. 3 \times 2; a_{21} = 3, a_{12} = -1; A\mathbf{v} = (8, 15, 13); only length-2 vectors are accepted.

3. Column-picture recomputation: 1.5\,(1,4,2,1) + 0.8\,(2,5,1,2) + 3.0\,(0,3,1,0) = (3.1,\ 19.0,\ 6.8,\ 3.1) mol — identical to the row-by-row totals of Exercise 2.

5. (a_1\ a_2\ a_3)\begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = a_1 b_1 + a_2 b_2 + a_3 b_3 — Definition 3.1.2 with one row, reproducing §2.3’s dot product as a 1 \times 1 product.

7. Entry i: \sum_j a_{ij}(u_j + v_j) = \sum_j a_{ij}u_j + \sum_j a_{ij}v_j, and \sum_j a_{ij}(cv_j) = c\sum_j a_{ij}v_j — distributivity of scalar arithmetic, row by row.

9. Solving the 3\times3 system exactly: feeds \approx (41.7,\ 29.2,\ 29.2) mol (i.e. \tfrac{125}{3}, \tfrac{175}{6}, \tfrac{175}{6}). The unpleasantness of finding it by trial is the point — §3.2’s inverse turns this into one multiplication.

§3.2 Matrix Algebra

1. AB = \begin{pmatrix} 4 & -2 \\ 6 & 0 \end{pmatrix}, BA = \begin{pmatrix} 2 & -2 \\ 4 & 2 \end{pmatrix}. The agreeing (1,2) entries (-2) are a numerical accident — perturb any entry of A and the coincidence evaporates.

3. \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix}: ad - bc = 0, no inverse (rows proportional — outputs fill only a line). \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}^{-1} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}. \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} is its own inverse: it swaps the two components, and swapping twice is doing nothing.

5. Diagonal matrices multiply diagonal-entry-by-diagonal-entry, and scalar multiplication commutes. Counterexample with a non-diagonal partner: D = \mathrm{diag}(1,2), M = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} give DM = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \neq \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix} = MD.

7. B = BI = B(AC) = (BA)C = IC = C — associativity does all the work, and “the” inverse is justified.

9. Good pair: c_X moves from 4.00 to 4.07 \times 10^{-5} M — about +1.7\%. Near-proportional pair (\det E' = 1.5 \times 10^6 versus 1.59 \times 10^8): the same \pm 0.01 in A_1 drives c_X from 4.0 to 5.9 \times 10^{-5} M — roughly +48\%, some thirty-fold worse. The foretelling scalar is the determinant: sensitivities scale like entries divided by \det, and E'’s determinant is two orders of magnitude smaller.

§3.3 Linear Transformations

1. R(270°) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}; reflection across y = x: \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}; stretch: \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}; R(120°) = \begin{pmatrix} -\tfrac12 & -\tfrac{\sqrt3}{2} \\ \tfrac{\sqrt3}{2} & -\tfrac12 \end{pmatrix}.

3. (1,0) \to (-\tfrac12, \tfrac{\sqrt3}{2}) \to (-\tfrac12, -\tfrac{\sqrt3}{2}) \to (1, 0): home in three steps, so R(120°)^3 = I. Any n-fold symmetry axis rotates by 360°/n; n applications complete a full turn, which is the identity — molecules cannot tell that anything happened, n times over.

5. R(60°) advances each carbon to its neighbor’s site — the set maps to itself. For 1,3,5-substitution the substituted and bare carbons alternate; the smallest rotation preserving each kind of site is 120°.

7. A rotation moves the plane rigidly: the parallelogram on \mathbf{u}, \mathbf{v} rotates as one object, so its diagonal (the sum) rotates to the diagonal of the rotated sides — R(\mathbf{u}+\mathbf{v}) = R\mathbf{u} + R\mathbf{v}. A scaled arrow rotates to the equally scaled rotated arrow. Both habits hold with no components in sight.

9. P^2 = P — once flattened onto the axis, projecting again changes nothing. No inverse: the y-component is destroyed (physically, information loss), and e.g. \mathbf{u} = (1, 0), \mathbf{v} = (1, 5) satisfy P\mathbf{u} = P\mathbf{v}, so no map can send the shared output back to both inputs — §3.2’s failed cancellation, realized geometrically.

11. (a) P(x, y) = (y, x) by direct multiplication. (b) \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} — its columns are the images of \hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}; on (1,2,3) it gives (3, 1, 2) ✓. (c) 3! = 6 — one matrix per ordering of three slots, the factorial of §1.3. (d) \det P = 0 \cdot 0 - 1 \cdot 1 = -1: a single swap reverses orientation, exactly as a reflection does — transpositions are the “odd” operations, and §3.4’s sign will keep the books.

§3.4 Determinants

1. \det = 2: invertible, orientation-preserving. \det = 0: singular. \det R(30°) = \cos^2 30° + \sin^2 30° = 1: invertible, preserving. \det = -1: invertible, orientation-reversing (it is the reflection across y = x).

3. AB = \begin{pmatrix} 4 & 6 \\ 10 & 12 \end{pmatrix}; \det A = -2, \det B = 6, \det(AB) = 48 - 60 = -12 = (-2)(6) ✓.

5. In their 2D in-plane representations: \det C_2 = \det \mathrm{diag}(-1, 1) = -1, \det \sigma_v = -1, \det R(120°) = +1. Physically |\det| = 1 always: a symmetry operation preserves all lengths, hence all areas and volumes — a scale factor other than 1 would visibly deform the molecule. (Subtlety worth savoring: the full 3D C_2 is a proper rotation with \det = +1; its restriction to the molecular plane looks like a reflection with \det = -1. The sign depends on the space you let the operation act on.)

7. Both columns pick up the factor c, each donating one power: \det(cA) = c^2 \det A (area picture: the spanned parallelogram stretches by c in two independent directions). The student’s 3\times3 error: halving all entries scales volume by (\tfrac12)^3 = \tfrac18, not \tfrac12.

9. \det E = 48 \times 10^6 - 48.8 \times 10^6 = -8 \times 10^5 — tiny against entries of order 10^4 \times 10^4. Measured (0.60, 0.40): \mathbf{c} = (5.0 \times 10^{-5},\ 0) M. Perturbed (0.61, 0.40): \mathbf{c} = (0,\ 1.0 \times 10^{-4}) M — a 1.7% absorbance change swapped the entire composition from all-X to all-Y. That is ill-conditioning: with near-parallel columns the inverse divides small differences by a small determinant, and noise is amplified without limit as \det \to 0.