- 1-1.1: Write the words used to read the number.22,356,027
- 1-1.2: Write the words used to read the number.106,357,291,582
- 1-1.3: Write the words used to read the number.730,531,968
- 1-1.4: Write the words used to read the number.21,000,017
- 1-1.5: Write the words used to read the number.523,800,007,190
- 1-1.6: Write the words used to read the number.713,205,538
- 1-1.7: Fourteen thousand, nine hundred eighty-five.
- 1-1.8: Thirty-two million, nine hundred forty-three thousand, six hundred ...
- 1-1.9: Seventeen billion, eight hundred three thousand, seventy-five.
- 1-1.10: Fifty million, six hundred twelve thousand, seventy-eight.
- 1-1.11: Three hundred six thousand, five hundred forty-one.
- 1-1.12: Three hundred million, seven hundred sixty thousand, five hundred t...
- 1-1.13: Round 483 to tens.
- 1-1.14: Round 3,762 to hundreds.
- 1-1.15: Round 298,596 to ten-thousands.
- 1-1.16: Round 57,802 to the first digit.
- 1-1.17: Cisco, the worlds largest Internet equipment maker, recorded earnin...
- 1-1.18: Net income at Levi Strauss, the worlds biggest maker of branded clo...
- 1-1.19: McDonalds produced 86,347,582 Big Macs. How many Big Macs were prod...
- 1-1.20: Oslo, Hong Kong, Tokyo, and New York City are the four most expensi...
- 1-1.21: According to Experian, a credit-reporting agency, the average debt ...
- 1-1.22: Experian reported that the average debt for people living in Missis...
- 1-1.23: Experian reported that people in the age range of 18 to 29 had the ...
- 1-1.24: Experian recently reported that people in the 50 to 69 age range ca...
Solutions for Chapter 1-1: PLACE VALUE AND OUR NUMBER SYSTEM
Full solutions for Business Math, | 9th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Invert A by row operations on [A I] to reach [I A-I].
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.