- 3-2.1: Add. 6.005 + 0.03 + 924 + 3.9
- 3-2.2: Add. 82 + 5,000.1 + 101.703
- 3-2.3: Add. $21.13 + $42.78 + $16.39
- 3-2.4: Add. $203.87 + $1,986.65 + $3,047.38 Subtract.
- 3-2.5: Subtract. 407.96 - 298.39
- 3-2.6: Subtract. 500.7 from 8,097.125
- 3-2.7: Subtract. $468.39 - $223.54
- 3-2.8: Subtract. $21.65 - $15.96
- 3-2.9: Subtract. $52,982.97 - $45,712.49
- 3-2.10: Subtract. $38,517 - $21,837.46
- 3-2.11: Multiply. 19.7
- 3-2.12: Multiply. 0.0321 * 10
- 3-2.13: Multiply. 73.7 * 0.02
- 3-2.14: Multiply. 43.7 * 1.23
- 3-2.15: Multiply. 5.03 * 0.073
- 3-2.16: Multiply. 642 * $12.98
- 3-2.17: Divide and round to the nearest hundredth if necessary 123.72 , 12
- 3-2.18: Divide and round to the nearest hundredth if necessary 35_589.06
- 3-2.19: Divide and round to the nearest hundredth if necessary 0.35_0.0084
- 3-2.20: Divide and round to the nearest hundredth if necessary 1,482.97 , 1.7
- 3-2.21: Kathy Mowers purchased items costing $14.97, $28.14, $19.52, and $2...
- 3-2.22: Jim Roznowski submitted a travel claim for meals, $138.42; hotel, $...
- 3-2.23: Joe Gallegos purchased a calculator for $12.48 and paid with a $20 ...
- 3-2.24: Martisha Jones purchased a jacket for $49.95 and a shirt for $18.50...
- 3-2.25: Laura Voight earns $8.43 per hour as a telemarketing employee. One ...
- 3-2.26: Cassie James works a 26-hour week at a part-time job while attendin...
- 3-2.27: Calculate the cost of 1,000 gallons of gasoline if it costs $2.47 p...
- 3-2.28: A buyer purchased 2,000 umbrellas for $4.62 each. What is the total...
- 3-2.29: All the employees in your department are splitting the cost of a ce...
- 3-2.30: AT&T offers a prepaid phone card for $5. The card provides 20 minut...
Solutions for Chapter 3-2: OPERATIONS WITH DECIMALS
Full solutions for Business Math, | 9th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
A sequence of steps intended to approach the desired solution.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Every v in V is orthogonal to every w in W.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·